Bifurcation currents in one-dimensional holomorphic dynamics

IntroductionThe aim of these lectures is the study of bifurcations within holomorphic familiesof polynomials or rational functions by mean of pluripotential-theoretic and ergodictools.The starting point of the subject is a discovery made by DeMarco [DeM2] whoshown that the bifurcation locus of any such family is the support of a (1, 1) closedpositive current which admits both the Lyapunov exponent function and the sum ofthe Green function evaluated on critical points as a global potential. This current,denoted T bif , is called the bifurcation current.In the recent years, several authors have investigated the geometry of the bifurcationlocus using the current T bif and its powers T bif ∧ T bif ∧ · · · ∧ T bif [DeM1], [DeM2],[BB1], [P], [DF], [Du], [BE], [BB2], [BB3], [G]. The approach followed by thesepapers enlights a certain stratification of the bifurcation locus which correspondsto the degree of self-intersection of T bif . The main results are exposed in thesenotes, they go from laminarity statements for certain regions of the bifucation locusto Hausdorff dimension estimates and includes precise density (or equidistribution)properties relative to various classes of specific parameters.We have not discussed bifurcation theory for families of endomorphisms of higherdimensional complex projective spaces but have mentionned, among the techniquespresented in these notes, those which also work in this more general context. Thisaspects appear in the papers [BB1], [BDM], [P] and in the survey [DS].We have tried to give a synthetic and self-contained presentation of the subject.In most cases, we have given complete and detailed proofs and, sometimes, havesubstantially simplify those available in the litterature. Although basics about ergodictheory are discussed in the first chapter, we have not treated the elementsof pluripotential theory. The other lectures delivered during this week will providemost of the needed knowledge. For that we also refer the reader to the appendixabout pluripotential theory available in Sibony’s [Sib] and Dinh and Sibony’s [DS]surveys or to Demailly’s book [Dem].3

Daigoro Isobe, Yutala ToiAt the member fracture phase in an element, the numerical integration point is shifted to theopposite end of the fractured section from the center of the element. Therefore, an imaginaryplastic hinge occurs at the fractured section according to the algorithm explained in thischapter. Generally, to release the resultant forces instantly is considered not the best way asthe longitudinal wave might occur in the process. In such cases, a gradual release technique ora technique considering artificial viscosity is used to stabilize the numerical calculation.However, in this paper, an instant release technique is used to make the algorithm simple.Rebounds and insertions to the ground, or contacts between elements are neglected in thisalgorithm.It is also to be noted that as elements and node points are still treated as a continuous modelin the calculation process, new imaginary node points for the fractured sections are needed tobe established at the post processing stage. The elements with the imaginary node points arethen visualized as rigid bars thereafter.3. DYNAMIC COLLAPSE ANALYSES OF REINFORCED CONCRETE BUILDINGIn this chapter, the explicit nonlinear code based on the U.L.F. with the ASI technique isimplemented into the existing finite element code, and the code has been applied to theexplosive demolition analysis of a five stories-five span reinforced concrete building. Also theseismic damage analysis has been conducted to the same model by the implicit nonlinear codebased on the U.L.F. with the ASI technique.3.1 Strength characteristics of reinforced concrete memberIn this section, strength characteristics of the reinforced concrete member used in theanalyses, is shown.Generally, many parameters such as shape, steel bar arrangement, material characteristics aswell as stress condition or stress history, contribute big influence to the strengthcharacteristics of a reinforced concrete member. Consequently, the limits of application rangeoccur with nature for proposal formulas and experimental data. In this paper, the formulaswhich are partially excluded for these restrictive matters are used 14 .A tri-linear type model, which has a crack point and an yield point, is employed. Theformulas used as crack and yield strengths are shown below:Flexural crack strength;M = 1.8 F ⋅ Z +cceNZAce( 8)Flexural yield strength;Shear crack strength;My0 ⎛ N.8 0.5 1 ⎟ ⎞a⎜t sfy ⋅ D + ND −⎝ bDFc⎠=⋅⎛ 1 N ⎞0.085QK150bDM Qd 1.7bjc = ⎜ + ⎟ cc⎝ ⎠+( 500 + F ) ( 10)( 9)5

4 Equidistribution towards the bifurcation current 514.1 Distribution of critically periodic parameters . . . . . . . . . . . . . . 514.1.1 Statement and a general strategy . . . . . . . . . . . . . . . . 514.1.2 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . 544.2 Distribution of rational maps with cycles of a given multiplier . . . . 564.2.1 The case of attracting cycles . . . . . . . . . . . . . . . . . . . 564.2.2 Averaging the multipliers . . . . . . . . . . . . . . . . . . . . . 594.2.3 The case of neutral cycles in polynomial families . . . . . . . . 634.3 Laminated structures in bifurcation loci . . . . . . . . . . . . . . . . . 654.3.1 Holomorphic motion of the Mandelbrot set . . . . . . . . . . . 654.3.2 Further laminarity statements for T bif . . . . . . . . . . . . . . 695 The bifurcation measure 715.1 A Monge-Ampère mass related with strong bifurcations . . . . . . . . 715.1.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 715.1.2 Some concrete families . . . . . . . . . . . . . . . . . . . . . . 735.2 Density statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.2.1 Misiurewicz parameters . . . . . . . . . . . . . . . . . . . . . . 755.2.2 Shishikura or hyperbolic parameters . . . . . . . . . . . . . . 775.2.3 Shishikura parameters with chosen multipliers . . . . . . . . . 795.3 The support of the bifurcation measure . . . . . . . . . . . . . . . . . 795.3.1 A transversality result . . . . . . . . . . . . . . . . . . . . . . 805.3.2 The bifurcation measure and strong-bifurcation loci . . . . . . 815.3.3 Hausdorff dimension estimates . . . . . . . . . . . . . . . . . . 846

Chapter 1Rational functions as ergodicdynamical systems1.1 Potential theoretic aspects1.1.1 The Fatou-Julia dichotomyA rational function f is a holomorphic map of the Riemann sphere to itself and maybe repesented as the ratio of two polynomialsf = a 0+a 1 z+a 2 z 2 +···+a d z db 0 +b 1 z+b 2 z 2 +···+b d z dwhere at least one of the coefficents a d and b d is not zero. The number d is thealgebraic degree of f. In the sequel we shall more likely speak of rational map. Sucha map may also be considered as a holomorphic ramified self-cover of the Riemannsphere whose topological degree is equal to d. Among these maps, polynomials areexactly those for which ∞ is totally invariant: f −1 {∞} = f{∞} = ∞.It may also be convenient to identify the Riemann sphere with the one-dimensionalcomplex projective space P 1 that is the quotient of C 2 \ {0} by the action z ↦→ u · zof C ∗ . Let us recall that the Fubini-Study form ω on P 1 satisfies π ∗ (ω) = dd c ln ‖ ‖where the norm is the euclidean one on C 2 .In this setting, the map f can be seen as induced on P 1 by a non-degenerate andd-homogenous map of C 2F (z 1 , z 2 ) := ( a 0 z d 2 + a 1 z 1 z d−12 + · · · + a d z d 1, b 0 z d 2 + b 1 z 1 z d−12 + · · · + b d z d 1through the canonical projection π : C 2 \ {0} → P 1The homogeneous map F is called a lift of f; all other lifts are proportional to F .A point around which a rational map f does not induce a local biholomorphismis called critical. The image of such a point is called a critical value. A degreed rational map has exactly (2d − 2) critical points counted with multiplicity. The7)

critical set of f is the collection of all critical points and is denoted C f .As for any self-map, we may study the dynamics of rational ones, that is tryingto understand the behaviour of the sequence of iteratesf n := f ◦ · · · ◦ f.The Fatou-Julia dynamical dichotomy consists in a splitting of P 1 into two disjointsubsets on which the dynamics of f is radically different. The Julia set ofa rational map f is the subset of P 1 on which the dynamics of f may drasticallychange under a small perturbation of initial conditions while the Fatou set is thecomplement of the Julia set.Definition 1.1.1 The Julia set J f and the Fatou set F f of a rational map f arerespectively defined by:J f := {z ∈ P 1 / (f n ) n is not equicontinuous near z}F f := P 1 \ J f .Both the Julia and the Fatou set are totally invariant: J f = f ( J f)= f−1 ( J f)and F f = f ( F f)= f−1 ( F f). In particular f induces two distinct dynamical systemson J f and F f .The dynamical system f : J f → J f is chaotical. However, as it results from theSullivan non-wandering theorem and the Fatou-Cremer classification, the dynamicsof a rational map is totally predictible on its Fatou set.The periodic orbits are called cycles and play a very important role in the understandingof the dynamics.Definition 1.1.2 A n-cycle is a set of n distinct points z 0 , z 1 , · · ·, z n−1 such thatf(z i ) = z i+1 for 0 ≤ i ≤ n − 2 and f(z n−1 ) = z 0 . One says that n is the exact periodof the cycle.Each point z i is fixed by f n . The multiplier of the cycle is the derivative of f nat some point z i of the cycle and computed in a local chart: ( χ ◦ f ◦ χ −1)′ (χ(z i )). Itis easy to see that this number depends only on the cycle and neither on the pointz i or the chart χ. By abuse we shall denote it (f n ) ′ (z i ).The local dynamic of f near a cycle is governed by the multiplier m. This leads tothe followingDefinition 1.1.3 The multiplier of a n-cycle is a complex number m which is equalto the deivative of f n computed in any local chart at any point of the cycle.When |m| > 1 the cycle is said repelling8

when |m| < 1 the cycle is said attractingwhen |m| = 1 the cycle is called neutral.Repelling cycles belongs to the Julia set and attracting one to the Fatou set. Forneutral cycles this depends in a very delicate way on the diophantine properties ofthe argument of m.The first fundamental result about Julia sets is the following.Theorem 1.1.4 Repelling cycles are dense in the Julia set.It is possible to give an elementary proof of that result using the Brody-Zalcmanrenormalization technique (see [BM]). We shall see later that repelling cycles actuallyequidistribute a measure whose support is exactly the Julia set.1.1.2 The Green measure of a rational mapOur goal is to endow the dynamical system f : J f → J f with an ergodic structurecapturing most of its chaotical nature. This is done by exhibiting an invariant measureµ f on J f which is of constant Jacobian. Such a measure was first constructedby Lyubich [L]. For our purpose it will be extremely important to use a potentialtheoreticapproach which goes back to Brolin [Br] for the case of polynomials. Wefollow here the presentation given by Dinh and Sibony in their survey [DS] whichalso covers mutatis mutandis the construction of Green currents for holomorphicendomorphisms of P k .The following Lemma is the key of the construction. It relies on the fundamentalfact thatd −1 f ⋆ ω = ω + dd c vfor some smooth function v on P 1 . This follows from a standard cohomology argumentor may be seen concretely by setting v := d −1 ‖F (z)‖ln for some lift F of f.‖z‖ dLemma 1.1.5 Up to some additive constant, there exists a unique continuous functiong on P 1 such that d −n f n⋆ ν → dd c g + ω for all positive measure ν which is givenby ν = ω + dd c u where u is continuous.Proof. Let us set g n := v + · · · + d −n+1 v ◦ f n−1 . One sees by induction thatd −n f n⋆ ν = ω + dd c g n + dd c (d −n u ◦ f). As the sequence (g n ) n is clearly uniformlyconverging, the conclusion follows by setting g := lim n g n .⊓⊔It might be useful to see how the function g can be obtained by using lifts.9

Lemma 1.1.6 Let F be a lift of a degree d rational map f. The sequence d −n ln ‖F n (z)‖converges uniformly on compact subsets of C 2 \ {0} to a function G F which satisfiesthe following invariance and homogeneity properties:i) G F ◦ F = dG Fii) G F (tz) = G F (z) + ln|t|, ∀t ∈ C.Moreover, G F − ln ‖ ‖ = g ◦ π where g is given by Lemma 1.1.5.Proof. Let us set G n (z) := d −n ln ‖F n (z)‖. As F is homogeneous and non-degeneratethere exists a constant M > 1 such that1M ‖z‖d ≤ ‖F (z)| ≤ M‖z‖ d .Thus 1 ‖F n (z)‖ d ≤ ‖F n+1 (z)‖ ≤ M‖F n (z)‖ d which, taking logarithms and dividingMby d n+1 yields |G n+1 (z)−G n (z)| ≤ ln M . This shows that Gd n+1 n is uniformly convergingto G F . The properties i) and ii) follows immediately from the definition of G F .(According to the proof of Lemma 1.1.5, g = lim n v + · · · + d −n+1 v ◦ f n−1)where v ◦ π = d −1 ‖F (z)‖ln .‖z‖ d To get the last assertion, it suffices to observe thatd −k v ◦ f k ◦ π = G k+1 − G k .⊓⊔The two above lemmas lead us to coin the followingDefinition 1.1.7 Let F be a lift of a degree d rational map f. The Green functionG F of F on C 2 is defined byG F := lim n d −n ln ‖F n (z)‖.The Green function of g F of F on P 1 is defined byG F − ln ‖ ‖ = g F ◦ π.We will sometimes use the notation g f instead of g F . The function g f is definedmodulo an additive constant.The function G F is p.s.h on C 2 with a unique pole at the origin.It is worth emphasize that both g and G F are uniform limits of smooth functions.In particular, these functions are continuous. One may actually prove more (see [DS]Proposition 1.2.3 or [BB1] Proposition 1.2):Proposition 1.1.8 The Green functions G F (z) and g F (z) are Hölder continuousin F and z.We are now ready to define the measure µ f and verify its first properties.10

Theorem 1.1.9 Let f be a degree d ≥ 2 rational map and g be a Green function off. Let µ f := ω + dd c g. Then µ f is a f-invariant probability measure whose supportis equal to J f . Moreover µ f has constant Jacobian: f ⋆ µ f = dµ f .Proof. As a weak limit of probability measures, µ f is a probability measure.We shall use Lemma 1.1.5 for showing that f ⋆ µ f = dµ f . By constructionv + d −1 g ◦ f = lim n v + d −1 g n ◦ f = lim n g n+1 = g and thus d −1 f ⋆ µ f = ω +dd c v + d −1 dd c (g ◦ f) = ω + dd c (v + d −1 g ◦ f) = µ f .The invariance property f ⋆ µ f = µ f follows immediately from f ⋆ µ f = dµ f byusing the fact that f ⋆ f ⋆ = d Id.Let us show that the support of µ f is equal to J f . If U ⊂ F f is open then f n⋆ ωis uniformly bounded on U and therefore µ f (U) = lim ∫ U d−n f n⋆ ω = 0, this showsthat Supp µ f ⊂ J f . Conversaly the identity f ⋆ µ f = dµ f implies that (Supp µ f ) cis invariant by f which, by Picard-Montel’s theorem, implies that (Supp µ f ) c ⊂ F f .⊓⊔It is sometimes useful to use the Green function G F for defining local potentialsof µ f .Proposition 1.1.10 Let f be a rational map and F be a lift. For any section σ ofthe canonical projection π defined on some open subset U of P 1 , the function G F ◦ σis a potential for µ f on U.Proof. On U one has dd c G F ◦ σ = dd c g F + dd c ln ‖σ‖ = dd c g F + ω = µ f .⊓⊔The measure µ f is the image by the canonical projection π : C 2 \ {0} → P 1 ofa Monge-Ampère measure associated to the Green function G F .Proposition 1.1.11 Let F be a lift of a degree d rational map f and G F a Greenfunction of F . The measure µ F := dd c G + F ∧ ddc G + Fis supported on the compact set{G F = 0} and satisfies F ⋆ µ F = d 2 µ F and π ⋆ µ F = µ f .This construction will be used only once in this text and we therefore skip itsproof. Observe that the support of µ F is contained in the boundary of the compactset K F := {G F ≤ 0} which is precisely the set of points z with bounded forwardorbits by F .The case of polynomials presents interesting features, in particular the Greenmeasure coincides with the harmonic measure of the filled-in Julia set.Proposition 1.1.12 let P be a degree d polynomial on C. The Green function g Pof P is the subharmonic function defined byg P := lim n d −n ln + |P n |11

and is a global potential of the Green measure of P .Proof. We may take F := ( )z2, d P ( z 1z 2), z2d as a lift of P . Then F n := ( )z2 dn P n ( z 1z 2), z2dnandG F (z 1 , 1) = 1 2 lim n d−n ln ( 1 + |P n (z 1 )| 2) = limnd −n ln + |P n (z 1 )|.The conclusion then follows from Proposition 1.1.10.⊓⊔1.2 Ergodic aspects1.2.1 Equidistribution towards the Green measure, mixingDefinition 1.2.1 Let (X, f, µ) be a dynamical system. One says that the measureµ is mixing if and only iffor any test functions ϕ and ψ.lim n∫X (ϕ ◦ f n ) ψ µ = ∫ X ϕ µ ∫ X ψ µThis means that the events {f n (x) ∈ A} and {x ∈ B} are asymptotically independantsfor any pair of Borel sets A, B.As we shall see, the constant Jacobian property implies that Green measures aremixing.Theorem 1.2.2 The Green measure µ f of any degree d rational map f is mixing.Proof. Let us set c ϕ := ∫ ϕ µ f and c ψ := ∫ ψ µ f where ϕ and ψ are two test functions.We may assume that c ϕ = 1.Since µ f and ϕµ f are two probability measures, there exists a smooth functionu ϕ on P 1 such that:ϕµ f = µ f + ∆u ϕ . (1.2.1)On the other hand, by the constant Jacobian property f ∗ µ f = µ f we have:d −n f n∗( (ϕ − c ϕ )µ f)=(ϕ ◦ f n − c ϕ)µf (1.2.2)Now, combining 1.2.1 and 1.2.2 we get:12

∫(ϕ ◦ f n )ψ µ f − ( ∫ )( ∫ ∫)ϕ µ f ψ µ f = (ϕ ◦ f n )ψ µ f − c ϕ c ψ =∫ψ ( ∫)ϕ ◦ f n − c ϕ µf = ψ d −n f n∗( ∫) (d(ϕ − c ϕ )µ f = −n f∗ n ψ) (ϕ − 1) µ f =∫ ∫∫∫(d −n f∗ n ψ) ∆u ϕ = ψ d −n f n∗ (∆u ϕ ) = ψ d −n ∆(u ϕ ◦ f n ) = d −n (u ϕ ◦ f n ) ∆ψ.This shows that lim n∫(ϕ ◦ f n )ψ µ f = ( ∫ ϕ µ f)( ∫ψ µf).⊓⊔It is not hard to show that a mixing measure is also ergodic.Definition 1.2.3 Let (X, f, µ) be a dynamical system. One says that the measureµ is ergodic if and only if all integrable f-invariant functions are constants.In particular this allows to use the classical Birkhoff ergodic theorem which saysthat time-averages along typical orbits coincide with the spatial-average:Theorem 1.2.4 Let (X, f, µ) be an ergodic dynamical system and ϕ ∈ L 1 (µ). Thenfor µ almost every x one has:∑ n−1k=0 ϕ(f k (z)) = ∫ ln ϕ µ Xlim n1nThe measure-theoretic counterpart of Fatou-Julia theorem 1.1.4 is the followingequidistribution result which has been first proved by Lyubich [L]. The content ofsubsection 1.3.2 will provide another proof which exploits the mixing property.Theorem 1.2.5 Let f be a rational map of degree d. Let R ⋆ n denote the set of nperiodic repelling points of f. Then d −n ∑ R ⋆ n δ z is weakly converging to µ f .Let us finally mention another classical equidistribution result. We refer to [DS]for a potential theoretic proof.Theorem 1.2.6 Let f be a degree d rational map. Thenlim n d −n ∑ {f n (x)=a} δ x = µ ffor any a ∈ P 1 which is not exceptional for f.We recall that a is exceptional for f if and only if a = ∞ and f is a polynomialor a ∈ {0, ∞} and f is of the form z ±d .Although this will not be used in this text, we mention that the Green measureµ f of any degree d rational map f is the unique measure of maximal entropy forf. This means that the entropy of µ f is maximal and, according to the variationalprinciple, equals ln d which is the value of the topological entropy of f.13

1.2.2 The natural extensionTo any ergodic dynamical system, it is possible to associate a new system which isinvertible and contains all the information of the original one. It is basically obtainedby considering the set of all complete orbits on which is acting a shift. Thisgeneral construction is the so-called natural extension of a dynamical system; hereis a formal definition.Definition 1.2.7 ( The natural ) extension of a dynamical system (X, f, µ) is the dynamicalsystem ̂X, ˆf, ˆµ wherêX := {ˆx := (x n ) n∈Z / x n ∈ X, f(x n ) = x n+1 }ˆf(ˆx) := (x n+1 ) n∈Zˆµ{(x n ) s.t. x 0 ∈ B} = µ(B).The canonical projection π 0 : ˆX → X is given by π0 (ˆx) = x 0 . One sets τ for ( ˆf) −1 .Let us stress that π 0 ◦ ˆf = f ◦ π 0 and (π 0 ) ∗ (ˆµ) = µ. The measure ˆµ inherits mostof the ergodic properties of µ.Proposition 1.2.8 The measure ˆµ is ergodic (resp.ergodic (rep. mixing).mixing) if and only if µ isWe refer the reader to the chapter 10 of [CFS] for this construction and its properties.A powerful way to control the behaviour of inverse branches along typical orbitsof the system (J f , f, µ f ) is to apply standard ergodic theory to its natural extension.This is what we shall do now. The first point is to observe that one may work withorbits avoiding the critical set of f. To this purpose one considerŝX reg = {ˆx ∈ Ĵf / x n /∈ C f ; ∀n ∈ Z}.As ˆµ is ˆf-invariant and µ does not give mass to points, one sees that ˆµ( ̂X reg ) = 1.Definition 1.2.9 Let ˆx ∈ ̂X reg and p ∈ Z. The injective map induced by f onsome neighbourhood of x p is denoted f xp . The inverse of f xp is defined on someneighbourhood of x p+1 and is denoted fx −1p. We then setf −nˆx:= f −1x −n◦ · · · ◦ f −1x −1.The map f −nˆxis called ”iterated inverse branch of f along ˆx and of depth n”.14

Then the conclusion follows by Birkhoff theorem since ‖F n (z)‖ stays away from0 and +∞ when z is in the support of µ F and π ⋆ µ F = µ f . ⊓⊔For any integer D the line bundle O P 1(D) over P 1 is the quotient of (C 2 \{0})×Cby the action of C ∗ defined by (z, x) ↦→ (uz, u D x). We denote by [z, x] the elementsof this quotient.The canonical metric on O P 1(D) may be written‖[z, x]‖ 0 = e −D ln ‖z‖ |x|.The homogenity property of G F allows us to define another metric on O P 1(D) bysetting‖[z, x]‖ GF= e −DG F (z) |x|.Let us underline that, according to Definition 1.1.7, ‖ · ‖ GF = e −Dg F‖ · ‖ 0 .The following Lemma will turn out to be extremely useful when we shall relatethe Lyapunov exponent with bifurcations.Lemma 1.3.4 Let f be a rational map of degree d ≥ 2 and F be one of its lifts. LetD := 2(d − 1) and Jac F be the holomorphic section of O P 1(D) induced by det F ′ .Then∫L(f) + ln d = ln ‖Jac F ‖ GF µ f .P 1Proof. The section Jac F is defined by Jac F (π(z)) := [z, det F ′ ] for any z ∈ C 2 \ {0}.Using Proposition 1.3.3, the fact that G F vanishes on the support of µ F and π ∗ µ F =µ f we get∫∫L(f) + ln d = ln | det F ′ | µ F = ln | det F ′ | µ F =C 2 G −1F∫∫({0})G −1F ({0}) ln ( e −DG F (z) | det F ′ | ) µ F =C 2 ln ‖Jac F ◦ π‖ GF µ F =∫P 1 ln ‖Jac F ‖ GF µ f .It is an important and not obvious fact that L(f) is stricly positive. It actuallyfollows from the Margulis-Ruelle inequality that L(f) ≥ 1 ln(d) where d is the degree2of f. We will presnt later a simple argument which shows that this bound is equalto ln d for polynomials. Zdunik and Mayer ([Z], [May]) have proved that the bound1ln(d) is taken if and only if the map f is a Lattès example. Let us summarize2these results in the following statement.17⊓⊔

Theorem 1.3.5 The Lyapunov exponent of a degree d rational map is always greaterthan 1 ln(d) and the equality occurs if and only if the map is a Lattès example.2We recall that a Lattès map is, by definition, induced on the Riemann spherefrom an expanding map on a complex torus by mean of some elliptic function. Werefer to the survey paper of Milnor [Mi2] for a detailed discussion of these maps.A remarkable consequence of the positivity of L(f) is that the iterated inversebranch f −nˆx(see definition 1.2.9) are approximately e −nL -Lipschiptz and are definedon a disc whose size only depends on ˆx.Proposition 1.3.6 There exists ɛ 0 > 0 and, for ɛ ∈]0, ɛ 0 ], two measurable functionsη ɛ : ̂Xreg →]0, 1] and S ɛ : ̂Xreg →]1, +∞] such that the maps f −nˆxare defined onD(x 0 , η ɛ (ˆx)) and Lip f −nˆx≤ S ɛ (ˆx)e −n(L−ɛ) for every n ∈ N and ˆµ-a.e. ˆx ∈ ̂X reg .Proof. We may assume that 0 < ɛ 0 < L −n. Since f3 ˆxassertion of Lemma 1.2.11 yields ln Lip f −nˆxergodic theorem we thus havelim sup 1 nln Lip f−nˆx≤ −L + ɛ 3≤ n ɛ 3 − ∑ n= fx −1−n◦ · · · ◦ fx −1−1, the thirdk=1 ln ρ(x −k). By Birkhofffor ˆµ-a.e. ˆx ∈ ̂X.Then there exists n 0 (ˆx) such(that Lip f −nˆx )≤ e −n(L−ɛ) for n ≥ n 0 (ˆx) and it sufficesto set S ɛ := max 0≤n≤n0 (ˆx) e n(L−ɛ) Lip f −nˆx to get the estimateLip f −nˆx≤ S ɛ (ˆx)e −n(L−ɛ) for every n ∈ N and ˆµ-a.e. ˆx ∈ ̂X reg .We now set η ɛ := αɛS ɛwhere α ɛ is the given by Proposition 1.2.10. Let us check byinduction on n ∈ N that f −nˆxis defined on D(x 0 , η ɛ (ˆx)) for ˆµ-a.e. ˆx ∈ ̂X and everyn ∈ N. Here we will use the fact that the function α ɛ is slow: α ɛ (τ(ˆx)) ≥ e −ɛ α ɛ (ˆx).Assume that f −nˆxis defined on D ( x 0 , η ɛ (ˆx) ) . Then, by our estimate on Lip f −nˆx, wehave( (f −nˆx D x0 , η ɛ (ˆx) )) ⊂ D ( x −n , e −n(L−ɛ) α ɛ (ˆx) ) .On the other hand, by Proposition 1.2.10, the branch fx −1−n−1is defined on the discD ( x −n , α ɛ (τ n+1 (ˆx)) ) which, as α ɛ is slow, contains D ( x −n , e −(n+1)ɛ α ɛ (ˆx) ) . Now,since 0 < ɛ 0 < L one has 3e−(n+1)ɛ ≥ e −n(L−ɛ) and thus f −(n+1)ˆxdefined on D ( x 0 , η ɛ (ˆx) ) .= f −1x −n−1◦ f −nˆxis⊓⊔1.3.2 Lyapunov exponent and multipliers of repelling cyclesThe following approximation property will play an important role in our study ofbifurcation currents. We would like to mention that Deroin and Dujardin haverecently used similar ideas to study the bifurcation in the context of kleinean groups(see [DD]).18

Theorem 1.3.7 Let f : P 1 → P 1 be a rational map of degree d ≥ 2 and L theLyapunov exponent of f with respect to its Green measure. Then:L = lim n d −n ∑ p∈R ∗ n1ln |(f n ) ′ (p)|nwhere R ∗ n := {p ∈ P1 / p has exact period n and |(f n ) ′ (p)| > 1}.Observe that the Lyapunov exponent lim k1k ln |(f k ) ′ (p)| of f along the orbit of apoint p is precisely equal to 1 n ln |(f n ) ′ (p)| when p is n periodic. The above Theoremthus shows that the Lyapunov exponent L of f is the limit, when n → +∞, of theaverages of Lyapunov exponents of repelling n-cycles.To establish the above Theorem, we will prove that the repelling cycles equidistributethe Green measure µ f in a somewhat constructive way and control the multipliersof the cycles which appear. For proving the equidistribution, we follow theapproach used by Briend-Duval [BD] in the context of endomorphisms of P k , thepositivity of the Lyapunov exponent plays a crucial role there. This strategy actuallyyields to a version of Theorem 1.3.7 for endomorphisms of P k ; this has beendone in [BDM]. Okuyama has given a different proof of Theorem 1.3.7 in [O], hisproof actually does not use the positivity of the Lyapunov exponent. The proof wepresent here is that of [Be] with a few more details.Proof. For the simplicity of notations we consider polynomials and therefore workon C with the euclidean metric. We shall denote D(x, r) the open disc centered atx ∈ C and radius r > 0. From now on, f is a degree d ≥ 2 polynomial whose Juliaset is denoted J and whose Green measure is denoted µ.We shall use the natural extension (see subsection 1.2.2) and exploit the positivityof L through Proposition 1.3.6. Let us add a few notations to those alreadyintroduced in Propositions 1.2.10 and 1.3.6 . Let 0 < ɛ 0 be given by Proposition1.2.10.For 0 < ɛ ≤ ɛ 0 and n, N ∈ N we set:̂X ɛ N := {ˆx ∈ ̂X / η ɛ (ˆx) ≥ 1 N and S ɛ(ˆx) ≤ N}ˆν ɛ N := 1 ̂Xɛ Nˆµν ɛ N := π ∗ˆν ɛ N .For 0 < ɛ ≤ L and n, N ∈ N we set:R ɛ n := {p ∈ C / f n (p) = p and |(f n ) ′ (p)| ≥ e n(L−ɛ) }µ ɛ n := d −n ∑ R ɛ n δ pR n := R L n = {p ∈ C / f n (p) = p and |(f n ) ′ (p)| ≥ 1}µ n := µ L n = d−n ∑ R nδ p .19

We have to show that µ L n → µ and L = lim ∑n d−nn R ln |(f n ) ′ (p)|.∗ nThe following Lemma reduces the problem to some estimates on Radon-Nikodymderivatives.Lemma 1.3.8 If any weak limit σ of ( )µ ɛ n for ɛ ∈]0, ɛ n 0[ satisfies dσ ≥ 1 for somedνNɛ′ɛ ′ > 0 and every N ∈ N then µ L n → µ and L = lim ∑n d−nn R ln |(f n ) ′ (p)|.∗ nProof.( )We start by showing that µ ɛ n → µ for any ɛ ∈ ]0, L]. Let σ be a weak limit ofµɛ . Since all the n n µɛ n are probability measures, it suffices to show that σ = µ.Assume first that 0 < ɛ < ɛ 0 . By assumption dσdν ɛ′N≥ 1 and therefore σ ≥ ν ɛ′N forevery N ∈ N. Letting N → +∞ one gets σ ≥ µ. This actually implies that σ = µsinceσ(J) ≤ lim sup n µ ɛ n (J) ≤ lim n dn +1d n= 1 = µ(J).We have shown that µ ɛ n → µ for 0 < ɛ < ɛ 0. Let us now assume that ɛ 02=: ɛ 1 ≤ ɛ.From µ ɛ n ≥ µ ɛ 1 n and µ ɛ 1 n → µ one gets σ ≥ µ. Just as before this implies that σ = µ.dWe now want to show that L = lim −nn n1ln |(f n ) ′ (p)|. For M > 0 one hasnµ ɛ n (J)(L − ɛ) ≤ d−n ∑ R ɛ nsince µ ɛ n → µ and µ n = µ L n → µ we get∑R ∗ n ln |(f n ) ′ (p)|. Let us set ϕ n (p) :=ϕ n (p) ≤ d ∑ ∫−n ϕ n (p) =R n∫≤JJln |f ′ |µ n ≤Max ( ln |f ′ |, −M ) µ n(L − ɛ) ≤ lim inf d −n ∑ R nϕ n (p) ≤ lim sup d −n ∑ R nϕ n (p) ≤∫JMax ( ln |f ′ |, −M ) µ.To obtain lim d ∑ −n R nϕ n (p) = L it suffices to make first M → +∞ and then ɛ → 0.Since there are less than 2nd n 2 periodic points whose period strictly divides n, onemay replace R n by Rn ∗ := {p ∈ P 1 / p has exact period n and |(f n ) ′ (p)| ≥ 1}. ⊓⊔Let us now finish the proof of Theorem 1.3.7. We assume here that 0 < ɛ < ɛ 02.Let â ∈ ̂X N ɛ and a := π(â). For every r > 0 we denote by D r the closed disc centeredat a of radius r. According to Lemma 1.3.8, it suffices to show that any weak limitσ of ( )µ 2ɛn satisfiesnσ(D r ′) ≥ ν ɛ N(D r ′), for any integer N and all 0 < r ′ < 1 N . (1.3.2)20

Let us pick r ′ < r < 1 N . We set ̂D r := π −1 (D r ) and :Ĉ n := {ˆx ∈ ̂D r ∩ ̂X reg ɛ N / f −nˆx(D r) ∩ D r ′ ≠ ∅}.Let also consider the collection S n of sets of the form f −nˆx(D r) where ˆx runs in Ĉn.As f −nˆxis an inverse branch on D r of the ramified cover f n , one sees that the setsof the collection S n are mutually disjoint.Let us momentarily admit the two following estimates:d −n( Card S n)≤ µ2ɛn (D r ) for n big enough (1.3.3)d −n( Card S n)µ(Dr ) ≥ ˆµ ( ˆf −n ( ̂D r ∩ ̂X ɛ reg N) ∩ ̂D r ′). (1.3.4)Combining 1.3.3 and 1.3.4 yields:ˆµ ( ˆf −n ( ̂D r ∩ ̂X ɛ reg N ) ∩ ̂D r ′which, by the mixing property of ˆµ, implies)≤ µ(Dr )µ 2ɛn (D r)ν ɛ N(D r )µ(D r ′) = ˆµ( ̂D r ∩ ̂X ɛ reg N)ˆµ( ̂D r ′) ≤ µ(D r )σ(D r )since µ(D r ′) > 0, one gets 1.3.2 by making r → r ′ .(Let us)now prove the estimate 1.3.3. We have to show that D r contains at leastCard Sn elements of R2ɛn when n is big enough. Here we shall use Proposition 1.3.6.For every ˆx ∈ Ĉn ⊂ ̂X reg ɛ N one has η ɛ(ˆx) ≥ 1 and S N ɛ(ˆx) ≤ N and thus the mapf −nˆxis defined on D r (r < 1 −n) and Diam fN ˆx(D r) ≤ 2r Lip f −nˆx≤ 2rS ɛ (ˆx)e −n(L−ɛ) ≤2rNe −n(L−ɛ) .As moreover f −nˆx (D r) meets D r ′, there exists n 0 , which depends only on ɛ, r and r ′ ,such that f −nˆx(D r) ⊂ D r for every ˆx ∈ Ĉn and n ≥ n 0 . Thus, by Brouwer theorem,f −nˆxhas a fixed point p n ∈ f −nˆx(D r) for every ˆx ∈ Ĉn and n ≥ n 0 . Since the elementsof S n are mutually disjoint sets, we have produced ( )Card S n fixed points of f n inD r for n ≥ n 0 . It remains to check that these fixed points belong to Rn 2ɛ.Thisactually follows immediately from the estimates on Lip f −nˆx. Indeed:|(f n ) ′ (p n )| = |(f −nˆx)′ (p n )| −1 ≥ ( )Lip f −n −1ˆx≥ N −1 e n(L−ɛ) ≥ e n(L−2ɛ)for n big enough.Finally we prove the estimate 1.3.4. Let us first observe thatπ ( ˆf −n ( ̂D r ∩ ̂X) ⋃reg N) ɛ ∩ ̂Dr ′ ⊂ f −nˆx (D r). (1.3.5)ˆx∈Ĉn21

This can be easily seen : if û ∈ ˆf −n ( ̂D r ∩ ̂X reg ɛ N )∩ ̂D r ′ then u 0 = π(û) ∈ D r ′∩f −nˆx (D r)where ˆx := ˆf n (û) ∈ ̂D r ∩ ̂X reg ɛ N .By the constant Jacobian property we have µ ( f −nˆx(D r) ) = d −n µ(D r ) and, since thesets f −nˆx(D r) of the collection S n are mutually disjoint, we obtainµ ( ⋃ˆx (D r) ) = ( )Card S n d −n µ(D r ). (1.3.6)ˆx∈Ĉn f −nCombining 1.3.5 with 1.3.6 yields 1.3.4:( )Card Sn d −n µ(D r ) ≥ µ [ π ( ˆf −n ( ̂D r ∩ ̂X) )]reg ɛ N ∩ ̂Dr ′ =ˆµ [ π −1 ◦ π ( ˆf −n ( ̂D r ∩ ̂X(reg N) ɛ ∩ ̂Dr ′)]≥ ˆµ ˆf −n ( ̂D r ∩ ̂X)reg N) ɛ ∩ ̂Dr ′ .⊓⊔22

Chapter 2Holomorphic familiesWe introduce here the main spaces in which we shall work in the subsequent chaptersand present some of their structural properties.2.1 Generalities2.1.1 Holomorphic families and the space Rat dLet us a sart with a formal definition.Definition 2.1.1 Let M be a complex manifold. A holomorphic mapf : M × P 1 → P 1such that all rational maps f λ := f(λ, ·) : P 1 → P 1 have the same degree d ≥ 2 iscalled holomorphic family of degree d rational maps parametrized by M. For short,any such family will be denoted (f λ ) λ∈M.Any degree d rational map f := a dz d +···+a 1 z+a 0b d z d +···+b 1 z+b 0is totally defined by the point[a d : · · · : a 0 : b d : · · · : b 0 ] in the projective space P 2d+1 . This allows to identify thespace Rat d of degree d rational maps with a Zariski dense open subset of P 2d+1 .We can be more precise by looking at the space of homogeneous polynomial mapsof C 2 which is identified to C 2d+2 by the correspondance(a d , · · ·, a 0 , b d : · · ·, b 0 ) ↦→( ∑di=1 a iz1 izd−i2 , ∑ )di=1 b iz1 izd−i2 .Indeed, Rat d is precisely the image by the canonical projection π : C 2d+2 → P 2d+1of the subspace H d of C 2d+2 consisting of non-degenerate polynomials. As H d is thecomplement in C 2d+2 of the projective variety defined by the vanishing of the resul-( ∑dtant Res(i=1 a iz1z i d−i2 , ∑ di=1 b iz1z i 2d−iis an (irreducible) algebraic hypersurface of P 2d+1 .), one sees that Rat d = P 2d+1 \ Σ d where Σ d23

Thus, when considering degree d polynomials, instead of working in P d we mayconsider the holomorphic family(Pc,a)(c,a)∈C d−1whose parameter space M is simply C d−1 . Using this parametrization, one mayexhibit a finite ramified cover π : C d−1 → P d (see [DF] Proposition 5.1).It will be crucial to consider the projective compactification P d−1 of C d−1 = M.This is why we wanted the expression of P c,a to be homogeneous in (c, a) and haveused the parameter a d instead of a. In this context, we shall denote by P ∞ theprojective space at infinity : P ∞ := {[c : a : 0] ; (c, a) ∈ C d−1 \ {0}}.2.1.3 Moduli spaces and the case of degree two rationalmapsThe group of Möbius transformations, which is isomorphic to P SL(2, C), acts byconjugation on the space Rat d of degree d rational maps. The dynamical propertiesof two conjugated rational maps are clearly equivalent and it is therefore natural towork within the quotient of Rat d resulting from this action.The moduli space Mod d is, by definition, the quotient of Rat d under the actionof P SL(2, C) by conjugation. We shall denote as follows the canonical projection:Π : Rat d −→ Mod df ↦−→ ¯fWe shall usually commit the abuse of language which consists in considering anelement of Mod d as a rational map. For instance, ” ¯f has a n-cycle of multiplier w”means that every element of ¯f posseses such a cycle. We shall also sometimes writef instead of ¯f.Although the action of P SL(2, C) is not free, it may be proven that Mod d is anormal quasi-projective variety [Sil].Remark 2.1.3 The following property is helpful for working in Mod d . Every elementf of Rat d belongs to a local submanifold T f whose dimension equals 2d − 2 andwhich is transversal to the orbit of f under the action of P SL(2, C). Moreover, T fis invariant under the action of the stabilizer Aut(f) of f which is a finite subgroupof P SL(2, C). Finally, Π ( T f)is a neighborhood of ¯f in Modd and Π induces abiholomorphism between T f /Aut(f) and Π ( T f).25

In his paper [Mi1], Milnor has given a particularly nice description of Mod 2which we will briefly present. The reader may also consult the fourth chapter ofbook of Silverman [Sil].A generic f ∈ Rat 2 has 3 fixed points with multipliers µ 1 , µ 2 , µ 3 . The symmetricfunctionsσ 1 := µ 1 + µ 2 + µ 3 , σ 2 := µ 1 µ 2 + µ 1 µ 3 + µ 2 µ 3 , σ 3 := µ 1 µ 2 µ 3are clearly well defined on Mod 2 and it follows from the holomorphic index formula∑ 11−µ i= 1 thatσ 3 − σ 1 + 2 = 0. (2.1.1)Milnor has actually shown that (σ 1 , σ 2 ) induces a good parametrization of Mod 2([Mi1]).Theorem 2.1.4 The map Mod 2 → C 2 defined by ¯f ↦→ (σ 1 , σ 2 ) is a biholomorphism.It will be extremely useful to consider the projective compactification of Mod 2obtained through the above Theorem:Mod 2 ∋ ¯f ↦−→ (σ 1 : σ 2 : 1) ∈ P 2whose corresponding line at infinity will be denoted by LL := {(σ 1 : σ 2 : 0); (σ 1 , σ 2 ) ∈ C 2 \ {0}}.It is important to stress that this compactification is actually natural in the sensethat the ”behaviour near L” captures a lot of dynamically meaningful information.2.2 The hypersurfaces Per n (w)We give some geometrical properties of dynamically defined subsets of the parameterspace which will play a central role in our study.2.2.1 Defining Per n (w) using dynatomic polynomialsFor any holomorphic family of rational maps, the following result describes preciselythe set of maps having a cycle of given period and multiplier.Theorem 2.2.1 Let f : M × P 1 → P 1 be a holomorphic family of degree d ≥ 2rational maps. Then for every integer n ∈ N ∗ there exists a holomorphic functionp n on M × C which is polynomial on C and such that:26

1- for any w ∈ C \ {1}, the function p n (λ, w) vanishes if and only if f λ has acycle of exact period n and multiplipler w2- p n (λ, 1) = 0 if and only if f λ has a cycle of exact period n and multiplier 1 ora cycle of exact period m whose multiplier is a primitive r th root of unity withr ≥ 2 and n = mr3- for every λ ∈ M, the degree N d (n) of p n (λ, ·) satisfies d −n N d (n) ∼ 1 n .This leads to the followingDefinition 2.2.2 Under the assumptions and notations of Theorem 2.2.1, one setsPer n (w) := {λ ∈ M/ p n (λ, w) = 0}for any integer n and any complex number w.According to Theorem 2.2.1, P er n (w) is (at least when w ≠ 1) the set of parametersλ for which f λ has a cycle of exact period n ad multiplier w. Moreover,Per n (w) is an hypersurface in the parameter space M or coincides with M. We alsostress that the estimate on the degree N d (n) of p n (λ, ·) will be important in some ofour applications.We now start to explain the construction of the functions p n . It clearly suffices totreat the case of the family Rat d and then set p n (λ, w) := p n (f λ , w) for any holomorphicfamily M ∋ λ ↦→ f λ ∈ Rat d . Our presentation borrows to the fourth chapterof the book of Silverman [Sil] and the paper [Mi1] of Milnor.We will consider polynomial families; to deal with the general case one mayadapt the proof by using lifts to C 2 . According to the discussion we had in subsection2.1.2, any degree d polynomial ϕ will be identified to a point in C d−1 .The key point is to associate to any integer n and any polynomial ϕ of degreed ≥ 2 a polynomial Φ ∗ ϕ,n whose roots, for a generic ϕ, are exactly the periodicpoints of ϕ with exact period n. Such polynomials are called dynatomic since theygeneralize cyclotomic ones, they are defined as follows.Definition 2.2.3 For a degree d polynomial ϕ and an integer n one setsΦ ϕ,n (z) := ϕ n (z) − z.The associated dynatomic polynomials are then defined by settingΦ ∗ ϕ,n(z) := ∏ (k|n Φϕ,k (z) ) µ( n k )where µ : N ∗ → {−1, 0, 1} is the classical Möbius function.27

It is clear that Φ ϕ,n is a polynomial whose roots are all periodic points of ϕ withexact period dividing n, and that Φ ∗ ϕ,n is a fraction whose roots and poles belong tothe same set. Actually Φ ∗ ϕ,n is still a polynomial but this is not at all obvious. Thiswill be shown by studying its valuation at any m-periodic point of ϕ for m|n anddeeply relies on the fact that the sum ∑ k|n µ( k ) vanishes if n > 1. We shall obtainna precise description of the roots of Φ ∗ ϕ,n :Theorem 2.2.4 Let ϕ be a polynomial of degree d ≥ 2. Then Φ ∗ ϕ,n is a polynomialwhose roots are the periodic points of ϕ with exact period m dividing n and multiplierw satisfying w r = 1 when 2 ≤ r := n . mThe proof of the above theorem will be given in the next subsection, for themoment we admit it and prove Theorem 2.2.1. Let us note that the degree ν d (n) ofΦ ∗ ϕ,n is given by ν d(n) = ∑ k|n dk µ( n k ) and is clearly equivalent to dn since |µ| ∈ {0, 1}and µ(1) = 1. We may also observe that (ϕ n ) ′ (z) = 1 for any root z of Φ ∗ ϕ,n whoseperiod strictly divides n.The construction of p n (ϕ, w) requires to understand the structure of the zeroset of (ϕ, z) ↦→ Φ ∗ ϕ,n(z). We recall that ϕ is seen as a point in C d−1 . Here are theinformations we need.Proposition 2.2.5 The set Per n := {(ϕ, z) / Φ ∗ ϕ,n (z) = 0} is an algebraic subset ofC d−1 ×C. The roots of Φ ∗ ϕ,n are simple and have exact period n when ϕ ∈ C d−1 \X nfor some proper algebraic subset X n of C d−1 .Proof. One sees on 2.2.3 that Φ ∗ ϕ,n (z) is rational in ϕ. On the other hand, Φ∗ ϕ,n (z)is locally bounded as it follows from the description of its roots given by Theorem2.2.4. Thus Φ ∗ ϕ,n(z) is actually polynomial in ϕ.Let us set ∆(ϕ) := ∏ i≠j (α i(ϕ) − α j (ϕ)) where the α i are the roots of Φ ∗ ϕ,n countedwith multiplicity. This is a well defined function which vanishes exactly when Φ ∗ ϕ,nhas a multiple root. This function is holomorphic outside its zero set and thereforeeverywhere by Rado’s theorem. Then {∆ = 0} is an analytic subset of C d−1 whichis proper since ϕ 0 := z d /∈ {∆ = 0}.Let Y n be the projection of P er n ∩ {(ϕ n ) ′ (z) = 1} onto C d−1 . By Remmert mappingtheorem Y n is an analytic subset of C d−1 . Using ϕ 0 again one sees that Y n ≠ C d−1 .One may take X n := {∆ = 0} ∪ Y n .⊓⊔Let Z(Φ ∗ ϕ,n) be the set of roots of Φ ∗ ϕ,n counted with multiplicity. If z ∈ Z(Φ ∗ ϕ,n)has exact period m with n = mr, we denote by w n (z) the r-th power of the multiplierof z (that is (ϕ n ) ′ (z)). As Theorem 2.2.4 tells us :a point z is periodic of exact period n and w n (z) ≠ 1 if and only ifz ∈ Z(Φ ∗ ϕ,n ) and w n(z) ≠ 1.28

Let us now consider the setsΛ ∗ n (ϕ) := {w n(z); z ∈ Z(Φ ∗ ϕ,n )}and let us denote by σ ∗(n)i (ϕ), 1 ≤ i ≤ ν d (n), the associated symmetric functions.The symmetric functions σ ∗(n)i are globally defined and continuous on C d−1 and,according to Proposition 2.2.5, are holomorphic outside X n . These functions aretherefore holomorphic on C d−1 . We setq n (ϕ, w) := ∏ ν d (n)i=0σ ∗(n)i (ϕ)(−w) ν d(n)−i .By construction q n (ϕ, w) = 0 if and only if w ∈ Λ ∗ n(ϕ) and q n is holomorphic in(ϕ, w) and polynomial in w. As Proposition 2.2.5 shows, the elements of Z(Φ ∗ ϕ,n )are cycles of exact period n and therefore each element of Λ ∗ n(ϕ) is repeated ntimes when ϕ /∈ X n . This means that there exists a polynomial p n (ϕ, ·) such thatq n (ϕ, ·) = (p n (ϕ, ·)) n when ϕ /∈ X n . As p n (ϕ, w) is holomorphic where it does notvanish one sees that p n extends to all C d−1 × C. In other words, p n may be definedby(p n (ϕ, w)) n := q n (ϕ, w) = ∏ ν d (n)i=0σ ∗(n)i (ϕ)(−w) νd(n)−i .The degree N d (n) of p n (λ, ·) is equal to 1 ν ∑n d(n) = 1 n k|n µ( n k )dk . In particulard −n N d (n) ∼ 1 .⊓⊔n2.2.2 The construction of dynatomic polynomialsWe aim here to prove Theorem 2.2.4. For this purpose, let us recall that the Möbiusfunction µ : N ∗ → {−1, 0, 1} enjoys the following fundamental property:∑ ( n)µ = 0 for any n ∈ N ∗ . (2.2.1)kk|nLet us also adopt a few more notations. The valuation of Φ ϕ,n(resp. Φ∗ϕ,n)atsome point z will be denoted a z (ϕ, n) ( resp. a ∗ z(ϕ, n)). The set of m-periodic pointsof ϕ will be denoted P er(ϕ, m).The following lemma summarizes elementary facts.Lemma 2.2.6 Let ψ be a polynomial and z ∈ P er(ψ, 1). Let λ := ψ ′ (z), then forq ≥ 2 one has:i) λ q ≠ 1 ⇒ a z (ψ, q) = a z (ψ, 1) = 1ii) λ ≠ 1 and λ q = 1 ⇒ a z (ψ, q) > a z (ψ, 1) = 129

iii) λ = 1 ⇒ a z (ψ, q) = a z (ψ, 1) ≥ 2.Proof. We may assume z = 0 and set ψ = λX + αX e + o(X e ). Then Φ ψ,q is equal to(λ q − 1) X + o(X) when (λ q − 1) ≠ 0 and to qαX e + o(X e ) if λ = 1. The assertionsi) to iii) then follow immediately. ⊓⊔We have to compute a ∗ z (ϕ, n) for z ∈ P er n(ϕ, m) and m|n. We denote by λ themultiplier of z (i.e. λ = (ϕ m ) ′ (z)) and set N := n . When λ is a root of unitymwe denote by r its order. Clearly, Theorem 2.2.4 will be proved if we establish thefollowing three facts:F1 : N = 1 ⇒ a ∗ z (ϕ, n) > 0F2 : N ≥ 2 and λ N ≠ 1 or λ = 1 ⇒ a ∗ z (ϕ, n) = 0F3 : N ≥ 2, λ N = 1 and λ ≠ 1 ⇒ a ∗ z(ϕ, n) ≥ 0 and a ∗ z(ϕ, n) > 0 iff r = N.Besides definitions, the following computation will use the obvious facts thata z (ϕ, k) = 0 when k ≠ qm for some q ∈ N ∗ and that ϕ qm = (ϕ q ) m :a ∗ z (ϕ, n) = ∑ ( n)µ a z (ϕ, k) = ∑ ( ) nµ a z (ϕ, qm) =k qmk|nqm|n∑( ) Nµ a z (ϕ, qm) = ∑ ( ) Nµ a z (ϕ m , q).q qq|Nq|NWe now proceed Fact by Fact, always starting with the above identity.F1) a ∗ z (ϕ, n) = µ(1)a z(ϕ n , 1) > 0.F2) Since λ q ≠ 1 when q|N although λ = 1, the assertions ( i) and iii) of Lemma∑ ( ))2.2.6 tell us that a z (ϕ m , q) = a z (ϕ m , 1). Then a ∗ z(ϕ, n) =q|N µ Naq z (ϕ m , 1)which, according to 2.2.1, equals 0.F3) When r is not dividing q then λ q ≠ 1 and, by the assertion i) of Lemma2.2.6 we have a z (ϕ m , q) = a z (ϕ m , 1). We may therefore write⎛a ∗ z (ϕ, n) = ⎝ ∑ ( ) ⎞ Nµ ⎠ a z (ϕ m , 1) + ∑ ( ) Nµ [a z (ϕ m , q) − a z (ϕ m , 1)] .qqq|Nq|N, r|qBy 2.2.1, the first term in the above expression vanishes and we geta ∗ z (ϕ, n) = ∑ ( ) N/rµ [a z (ϕ m , rk) − a z (ϕ m , 1)] =kk| N r∑( ) N/rµ [a z (ϕ mr , k) − a z (ϕ m , 1)] = a ∗kz(ϕ mr , N r ) − a z(ϕ m , 1) ∑ ( ) N/rµ .kk| N k| N rr30

We finally consider two subcases.( )N/rkIf N ≠ r then, by 2.2.1, ∑ k|µ = 0 and thus a ∗ N z (ϕ, n) = a∗ z (ϕmr , N ).rrSince z is a fixed point of ϕ mr whose multiplier equals λ r = 1, Fact F2 shows thata ∗ z(ϕ mr , N ) = 0. rIf N = r we get a ∗ z (ϕ, n) = a∗ z (ϕmr , 1) − a z (ϕ m , 1) = a z (ϕ mr , 1) − a z (ϕ m , 1). Sincez is a fixed point of ϕ m whose multiplier λ satisfies λ r = λ N = 1 and λ ≠ 1, theassertion ii) of Lemma 2.2.6 shows that this quantity is strictly positive. ⊓⊔Remark 2.2.7 It follows easily from the above construction that the growth ofp n (λ, w) is polynomial in λ when λ ∈ C d−1 . This shows that p n (λ, w) is actually apolynomial function on C d−1 × C.2.2.3 In the moduli space of degree two rational mapsTheorem 2.2.8 The map Mod 2 → C 2 defined by ¯f ↦→ (σ 1 , σ 2 ) is a biholomorphism.Using this identification, p n (λ, w) is a polynomial on C 2 × C. Moreover, for everyfixed w ∈ C, the degree of p n (·, w) is equal to ν 2(n)which is the number of hyperbolic2components of period n in the Mandelbrot set.Let us recall the projective compactification Mod 2 ∋ ¯f ↦−→ (σ 1 : σ 2 : 1) ∈ P 2whose corresponding line at infinity L = {(σ 1 : σ 2 : 0); (σ 1 , σ 2 ) ∈ C 2 \ {0}}.Any P er n (w) may be seen as a curve in P 2 . We shall use the following factswhich are also due to Milnor ([Mi1]).Proposition 2.2.9 1) For all w ∈ C the curve P er 1 (w) is actually a line whoseequation in C 2 is (w 2 + 1)λ 1 − wλ 2 − (w 3 + 2) = 0 and whose point at infinityis (w : w 2 + 1 : 0). In particular, P er 1 (0) = {λ 1 = 2} is the line of quadraticpolynomials, its point at infinity is (0 : 1 : 0).2) For n > 1 and w ∈ C the points at infinity of the curves P er n (w) are of theform (u : u 2 + 1 : 0) with u q = 1 and q ≤ n.The following Proposition, also du to Milnor (see Theorem 4.2 in [Mi1]), impliesthat the curves P er n (w) = {p n (·, w) = 0} have no multiplicity.Proposition 2.2.10 Let N 2 (n) := Card (P er n (0) ∩ P er 1 (0)) be the number of hyperboliccomponents of period n in the Mandelbrot set. Then N 2 (n) = ν 2(n)where2ν 2 (n) is defined inductively by ν 2 (1) = 2 and 2 n = ∑ k|n ν 2(k). Moreover, for anyw ∈ ∆ and any η ∈ ∆ we have Deg p n (·, w) = N 2 (n) = Card (P er n (w) ∩ P er 1 (η)) .Working with the same compactification, Epstein [Eps1] has proved the boundednessof certain hyperbolic components of Mod 2 :31

Theorem 2.2.11 Let H be a hyperbolic component of Mod 2 whose elements admittwo distinct attracting cycles. If neither attractor is a fixed point then H is relativelycompact in Mod 22.3 The connectedness locus in polynomial families2.3.1 Connected and disconnected Julia sets of polynomialsAmong rational functions, polynomials are characterized by the fact that ∞ is atotally invariant critical point. For any polynomial P the super-attractive fixedpoint ∞ determines a basin of attractionB P (∞) := {z ∈ C s.t. lim n P n (z) = ∞}.This basin is always connected and its boundary is precisely the Julia set J P ofP . The complement of B P (∞) is called the filled-in Julia set of P .Another nice feature of polynomials is the possibility to define a Green functiong P by settingg P (z) := lim n1deg(P ) n ln + |P n (z)|.The Green function g P is a subharmonic function on the complex plane which vanishesexactly on the filled-in Julia set of P .Any degree d polynomial P is locally conjugated at infinity with the polynomialz d . This means that there exists a local change of coordinates ϕ P (which is calledBöttcher function) such that ϕ P ◦ P = (ϕ P ) d on a neighbourhood of ∞. It isimportant to stress the following relation between the Böttcher and Green functions:ln |ϕ P | = g P .The only obstruction to the extension of the Böttcher function ϕ P to the fullbasin B P (∞) is the presence of other critical points than ∞ in B P (∞). This leadsto the following important result:Theorem 2.3.1 For any polynomial P of degree d ≥ 2 the folllowing conditions areequivalent:i) B P (∞) is simply connectedii) J P is connectediii) C P ∩ B P (∞) = {∞}32

iv) P is conformally conjugated to z d on B P (∞).The above theorem gives a nice characterization of polynomials having a connectedJulia set. Let us apply it to the quadratic family (z 2 + a) a∈C. The Juliaset J a of P a := z 2 + a is connected if and only if the orbit of the critical point 0 isbounded. In other words, the set of parameters a for which J a is connected is thefamousDefinition 2.3.2 Mandelbrot set. Let P a denote the quadratic polynomial z 2 +a.The Mandelbrot set M is defined byM := {a ∈ C s.t. sup n |P n a (0)| < ∞}.The Mandelbrot set is thus the connectedness locus of the quadratic family. It isnot difficult to show that M is compact. The compacity of the connectedness locusin the polynomial families of degree d ≥ 3 is a much more delicate question whichhas been solved by Branner and Hubbard [BH]. We shall treat it in the two nextsubsections and also present a somewhat more precise result which will turn out tobe very useful later.2.3.2 Polynomials with a bounded critical orbitWe work here with the parametrization ( P c,a)(c,a)∈C d−1 of P d and will use the projectivecompactification P d−1 introduced in the subsection 2.1.2.We aim to show that the subset of parameters (c, a) for which the polynomial P c,ahas at least one bounded critical orbit can only cluster on certain hypersurfaces ofP ∞ . The ideas here are essentially those used by Branner and Hubbard for provingthe compactness of the connectedness locus (see [BH] Chapter 1, section 3) but wealso borrow from the paper ([DF]) of Dujardin and Favre.We shall use the followingDefinition 2.3.3 The notations are those introduced in subsection 2.1.2. For every0 ≤ i ≤ d − 2, the hypersurface Γ i of P ∞ is defined by:Γ i := {[c : a : 0]/ α i (c, a) = 0}where α i is the homogeneous polynomial given by:α i (c, a) := P c,a (c i ) = 1 d−1d cd i + ∑j=2(−1) d−jσ d−j (c)c j ij+ ad .We denote by B i the set of parameters (c, a) for which the critical point c i of P c,ahas a bounded forward orbit (recall that c 0 = 0):33

The key point is the followingB i := {(c, a) ∈ C d−1 s.t. sup n |P n c,a (c i)| < ∞}.Lemma 2.3.4 The intersection Γ 0 ∩ Γ 1 ∩ · · · ∩ Γ d−2 is empty and Γ i1 ∩ · · · ∩ Γ ikcodimension k in P ∞ if 0 ≤ i 1 < ·· < i k ≤ d − 2.hasProof. A simple degree argument shows that P c,a (0) = P c,a (c 1 ) = ··· = P c,a (c d−2 ) = 0implies that c 1 = · · · = c d−2 = a = 0. Thus Γ 0 ∩ Γ 1 ∩ · · · ∩ Γ d−2 = ∅. Then theconclusion follows from Bezout’s theorem.⊓⊔Since the connectedness locus coincides with ∩ 0≤i≤d−2 B i , the announced resultcan be stated as follows.Theorem 2.3.5 For every 0 ≤ i ≤ d − 2, the cluster set of B i in P ∞ is containedin Γ i . In particular, the connectedness locus is compact in C d−1 .Let us mention the following interesting consequence. We recall that, accordingto Remark 2.2.7, the sets P er m (η) may be seen as algebraic subsets of the projectivespace P d−1 .Corollary 2.3.6 If 1 ≤ k ≤ d − 1, m 1 < m 2 < · · · < m k and sup 1≤i≤k |η i | < 1 thenPer m1 (η 1 )∩···∩Per mk (η k ) is an algebraic subset of codimension k whose intersectionwith C d−1 is not empty.Proof. By Bezout’s theorem, Per m1 (η 1 ) ∩ · · · ∩ Per mk (η k ) is a non-empty algebraicsubset of P d−1 whose dimension is bigger than (d − 1 − k).Any cycle of attracting basins capture a critical orbit. Therefore, Theorem 2.3.5implies that the intersection of P ∞ with Per m1 (η 1 ) ∩ · · · ∩ Per mk (η k ) is contained insome Γ i1 ∩···∩Γ ik since the m i are mutually distinct and the |η i | strictly smaller than1. Then, according to Lemma 2.3.4, P ∞ ∩ Per m1 (η 1 ) ∩ · · · ∩ Per mk (η k ) has codimensionk in P ∞ . The conclusion now follows from obvious dimension considerations. ⊓⊔As the Green function g c,a of the polynomial P c,a is defined byone sees thatg c,a (z) := lim n d −n ln + |P n c,a(z)|B i = {(c, a) ∈ C d−1 s.t. g c,a (c i ) = 0}.This is why the proof of Theorem 2.3.5 will rely on estimates on the Green functionsand, more precisely, on the following result.Proposition 2.3.7 Let g c,a be the Green function of P c,a and G be the functiondefined on C d−1 by: G(c, a) := max{g c,a (c k ); 0 ≤ k ≤ d − 2}. Let δ :=Then the following estimates occur:34∑ d−2k=0 c kd−1.

1) G(c, a) ≤ ln max{|a|, |c k |} + O(1) if max{|a|, |c k |} ≥ 12) max{g c,a (z), G(c, a)} ≥ ln |z − δ| − ln 4.Let us first show how Theorem 2.3.5 may be deduced from Proposition 2.3.7.Proof of Theorem 2.3.5. Let ‖(c, a)‖ ∞ := max ( |a|, |c k | ) . We simply have to check(that α (c,a))i ‖(c,a)‖ ∞tends to 0 when ‖(c, a)‖∞ tends to +∞ and g c,a (c i ) stays equal to0. As P c,a (c i ) = α i (c, a) and g c,a (c i ) = 0, the estimates given by Proposition 2.3.7yield:ln ‖(c, a)‖ ∞ + O(1) ≥ max ( dg c,a (c i ), G(c, a) ) = max ( g c,a ◦ P c,a (c i ), G(c, a) ) ≥since α i is d-homogeneous we then get:(1 − d) ln ‖(c, a)‖ ∞ + O(1) ≥ ln 1 4 |α ( (c, a) )i−‖(c, a)‖ ∞and the conclusion follows sinceδ‖(c,a)‖ d ∞Let us end this subsection by giving a≥ ln 1 4 |α i(c, a) − δ|δ|‖(c, a)‖ d ∞tends to 0 when ‖(c, a)‖ ∞ tends to +∞. ⊓⊔Proof of Proposition 2.3.7. The first estimate is a standard consequence of theuniform growth of P c,a at infinity. Let us however prove it with care. We will setA := ‖(c, a)‖ ∞ and M A (z) := max(A, |z|). From|P c,a (z)| ≤ 1|z|d( 1 + d max{ |σ d−j(c)|, |a|d } )d j|z| d−j |z| dwe get |P c,a (z)| ≤ C d |z| d for |z| ≥ A where the constant C d only depends on d. Wemay assume that C d ≥ 1. By the maximum modulus principle this yieldsIt is easy to check that|P c,a (z)| ≤ C d M A (z) d .M A (Cz) ≤ CM A (z) if C ≥ 1M A(MA (z) N) = M A (z) N if A ≥ 1.From now on we shall assume that A ≥ 1. By induction one getswhich implies|Pc,a(z)| n ≤ C 1+d+···+dn−1dM A (z) dng c,a (z) ≤ ln C dd−1 + ln max ( ‖(c, a)‖ ∞ , |z| ) if ‖(c, a)‖ ∞ ≥ 135

and in particularG(c, a) ≤ ln C dd−1 + ln ‖(c, a)‖ ∞ if ‖(c, a)‖ ∞ ≥ 1.The second estimate is really more subtle. It exploits the fact that the Greenfunction g P coincides with the log-modulus of the Böttcher coordinate function ϕ Pand relies on a sharp control of the distorsions of this holomorphic function.The Böttcher coordinate function is a univalent function ϕ c,a : {g c,a > G(c, a)} →C such that ϕ c,a ◦ P c,a = ϕ d c,a. It is easy to check that ln |ϕ c,a | = g c,a where it makessense and that ϕ c,a (z) = z − δ + O( 1 z ) where δ := σ 1(c)d−1= ∑ c kd−1 .One thus sees that ϕ c,a : {g c,a > G(c, a)} → C \ D ( 0, e G(c,a)) is a univalent mapwhose inverse ψ c,a satisfies ψ c,a (z) = z + δ + O( 1 ) at infinity. We shall now applyzthe following result, which is a version of the Koebe 1 -theorem (see [BH], Corollary43.3), to ψ c,a .Theorem 2.3.8 If F : Ĉ \ D r → Ĉ is holomorphic and injective andthen C \ D 2r ⊂ F ( C \ D r).F (z) = z + ∑ ∞ a nn=1 z n , z ∈ C \ D rPick z ∈ C and set r := 2 max ( e gc,a(z) , e G(c,a)) ( ). Then z /∈ ψ c,a C \ Dr sinceotherwise we would have e gc,a (z) = |ϕ c,a (z)| > r ≥ 2e gc,a (z).Thus, according to the above distorsion theorem, z /∈ C \ D ( δ, 2r ) . In other words|z − δ| ≤ 2r = 4 max ( e gc,a(z) , e G(c,a)) and the desired estimate follows by takinglogarithms.⊓⊔36

Chapter 3The bifurcation currentWe consider in this chapter an arbitrary holomorphic family of degree d rationalmaps (f λ ) λ∈Mwith marked critical points (see 2.1.2). The Julia set of f λ wil bedenoted J λ , its critical set C λ .Our first aim is to describe necessary and sufficient conditions for the Julia setsJ λ to move continuously (and actually holomorphically) with the parameter λ. Theparameters around which such a motion does exist are called stable. We will showthat the set of stable parameters is dense in the parameter space; this is the essenceof the Mañé-Sad-Sullivan theory. We will then exhibit a closed positive (1, 1)-currenton the parameter space whose support is precisely the complement of the stabiltylocus; this is the bifurcation current.3.1 Stability versus bifurcation3.1.1 Motion of repelling cycles and Julia setsAs Julia sets coincide with the closure of the sets of repelling cycles (see Theorems1.1.4 and 1.2.5), it is natural to investigate how J λ varies with λ through theparametrizations of such cycles.Assume that f λ0 has a repelling n-cycle {z 0 , f λ0 (z 0 ), · · ·, f n−1λ 0(z 0 )}. Then, theimplicit function theorem, applied to the equation fλ n(z) − z = 0 at (λ 0, z 0 ) showsthat there exists a holomorphic map h λ (z 0 ) : U 0 → P 1 such that h λ0 (z 0 ) = z 0 andh λ (z 0 ) is a n-periodic repelling point of f λ for all λ ∈ U 0 . Moreover, h λ (·) can beextended to the full cycle so that f λ ◦ h λ = h λ ◦ f λ0 .This shows that every repelling n-cycle of f λ0 moves holomorphically on some neighbourhoodof λ 0 .These observations lead to the following formal definition.37

Definition 3.1.1 Let us denote by R λ,n the set of repelling n-cycles of f λ . Let Ωbe a neighbourhood of λ 0 in M. One says that R λ0 ,n moves holomorphically on Ω ifthere exists a maph : Ω × R λ0 ,n ∋ (λ, z) ↦→ h λ (z) ∈ R λ,nwhich depends holomorphically on λ and satisfies h λ0 = Id, f λ ◦ h λ = h λ ◦ f λ0 .More generally, the holomorphic motion of an arbitrary subset of the Riemannsphere is defined in the following way.Definition 3.1.2 Let E be subset of the Riemann sphere and Ω be a complex manifold.Let λ 0 ∈ Ω. An holomorphic motion of E over Ω centered at λ 0 is a mapwhich satisfies the following properties:h : Ω × E ∋ (λ, z) ↦→ h λ (z) ∈ Ĉi) h λ0 = Id| Eii) E ∋ z ↦→ h λ (z) is one-to-one for every λ ∈ Ωiii) Ω ∋ λ ↦→ h λ (z) is holomorphic for every z ∈ E.The interest of holomorphic motions relies on the fact that any holomorphicmotion of a set E extends to the closure of E. This is a quite simple consequenceof Picard-Montel theorem.Lemma 3.1.3 (basic λ-lemma) Let E ⊂ Ĉ be a subset of the Riemann sphereand σ : E × Ω ∋ (z, t) ↦→ σ(z, t) ∈ Ĉ be a holomorphic motion of E over Ω. Thenσ extends to a holomorphic motion ˜σ of E over Ω. Moreover ˜σ is continuous onE × Ω.As J λ is the closure of the set of repelling cycles of f λ , this lemma implies thatthe Julia set J λ0 moves holomorphically over a neighbourhood V λ0 of λ 0 in M assoon as all repelling cycles of f λ0 move holomorphically on V λ0 . Moreover, the holomorphicmotion obtained in this way clearly conjugates the dynamics: h λ (J λ0 ) = J λand f λ ◦ h λ = h λ ◦ f λ0 on J λ0 .The above arguments lead to the following basic observation.Lemma 3.1.4 If there exists a neighbourhood Ω of λ 0 in the parameter space Msuch that, for all sufficebtly big n , R λ0 ,n moves holomorphically on Ω, then thereexists a holomorphic motion h λ : J λ0 → J λ which conjugates the dynamics.38

Let us mention here that there exists a much stronger version of the λ-lemma,which is due to Slodkowski and shows that the holomorphic motion actually extendsas a quasi-conformal transformation of the full Riemann sphere (see Theorem4.3.4). In particular, under the assumption of the above lemma, f λ and f λ0 arequasi-conformally conjugated when λ ∈ Ω.We may now define the set of stable parameters and its complement; the bifurcationlocus.Definition 3.1.5 Let (f λ ) λ∈Mbe a holomorphic family of degree d rational maps.The stable set S is the set of parameters λ 0 ∈ M for which there exists a neighbourhoodΩ of λ 0 and a holomorphic motion h λ of J λ0 over Ω centered at λ 0 and suchthat f λ ◦ h λ = h λ ◦ f λ0 on J λ0 .The bifurcation locus Bif is the complement M \ S.By definition, S is an open subset of M but it is however not yet clear that it isnot empty. We shall actually show that S is dense in M. To this purpose we willprove that the stability is characterized by the stability of the critical orbits. Thenext subsection will be devoted to this simple but remarkable fact.3.1.2 Stability of critical orbitsLet us start by explaining why bifurcations are related with the instability of criticalorbits.As we saw in Lemma 3.1.4, a parameter λ 0 belongs to the bifurcation locus if forany neighbourhood Ω of λ 0 in the parameter space M there exists n 0 ≥ 0 for whichR λ0 ,n 0does not move holomorphically on Ω.It is not very difficult to see that this forces one of the repelling n 0 -cycles of f λ0 ,say R λ0 , to become neutral and then attracting for a certain value λ 1 ∈ Ω. Nowcomes the crucial point. A classical result asserts that the basin of attraction ofany attracting cycle of a rational map contains a critical point (see [BM] ThéorèmeII.5). Thus, one of the critical orbit f k λ (c i(λ)) is uniformly converging to R λ on aneighbourhood of λ 1 . Then the sequence f k λ (c i(λ)) cannot be normal on Ω sinceotherwise, by Hurwitz lemma, it should converge uniformly to R λ which is repellingfor λ close to λ 0 . This arguments show that the bifurcation locus is contained in theset of parameters around which the post-critical set does not move continuously.This is an extremely important observation because it will allow us to detect bifurcationsby considering only the critical orbits. It leads to the following definitions.Definition 3.1.6 Let (f λ ) λ∈Mbe a holomorphic family of degree d rational maps.A marked critical point c(λ) is said to be passive at λ 0 if the sequence ( f n λ (c(λ)) ) n39

is normal on some neighbourhood of λ 0 . If c(λ) is not passive at λ 0 one says that itis active. The activity locus of c(λ) is the set of parameters at which c(λ) is active.The key result may now be given.Lemma 3.1.7 In a holomorphic family with marked critical points the bifurcationlocus coincide with the union of the activity loci of the critical points.Proof. According to our previous arguments, the bifurcation locus is contained inthe union of the activity loci. It remains to show that, for any marked critical pointc(λ), the sequence ( fλ n (c(λ)) ) is normal on S. Assume that λ n 0 ∈ S. As J λ0 is aperfect compact set, we may find three distinct points a 1 , a 2 , a 3 on J λ0 which areavoided by the orbit of c(λ 0 ) . Since the holomorphic motion h λ of J λ0 conjugatesthe dynamics, the orbit of c(λ) avoids {h λ (a j ); 1 ≤ j ≤ 3} for all λ in a smallneighbourhood of λ 0 . The conclusion then follows from Picard-Montel’s Theorem.⊓⊔The following lemma is quite useful.Lemma 3.1.8 If λ 0 belongs to the activity locus of some marked critical point c(λ)then there exists a sequence of parameters λ k → λ 0 such that c(λ k ) belongs to somesuper-attracting cycle of f λk or is strictly preperiodic to some repelling cycle of f λk .Proof. Since c(λ) is active at λ 0 it cannot be persistently fixed. After an arbitrarilysmall perturbation of λ 0 we may asume that there exists holomorphicmaps c −2 (λ), c −1 (λ) near λ 0 such that f λ (c −2 (λ)) = c −1 (λ), f λ (c −1 (λ)) = c(λ)(and Card {c −2 (λ), c −1 (λ), c(λ)} = 3. By Picard-Montel Theorem, the sequencefnλ (c(λ)) ) cannot avoid {c n −2(λ), c −1 (λ), c(λ)} on any neighbourhood of λ 0 .A similar argument using Picard-Montel Theorem and a repelling cycle of periodn 0 ≥ 3 shows that c(λ) becomes strictly preperiodic for λ arbitrarily close to λ 0 . ⊓⊔We are now ready to state and prove Mañé-Sad-Sullivan theorem.Theorem 3.1.9 Let (f λ ) λ∈Mbe a holomorphic family of degree d rational maps withmarked critical points {c 1 (λ), · · ·, c 2d−2 (λ)}.A parameter λ 0 is stable if one of the following equivalent conditions is satisfied.1) J λ0 moves holomorphically around λ 0 (see definition 3.1.5)2) the critical points are passive at λ 0 (see definition 3.1.6)3) f λ has no unpersistent neutral cycles for λ sufficently close to λ 0 .The set S of stable parameters is dense in M.40

Proof. The equivalence between 1) and 2) is given by Lemma 3.1.7. Similar argumentsmay allow to show that 3) is an equivalent statement (we will give analternative proof of that later). It remains to show that S is dense in M. Accordingto Lemma 3.1.8 we may perturb λ 0 and assume that f λ0 has a superattracting cycleof period bigger than 3 which persists, as an attracting cycle, for λ close enoughto λ 0 . If λ 0 is still active, Picard-Montel’s Theorem shows that a new perturbationguarantees that a critical point falls in the attracting cycle and, therefore, becomespassive. Since the number of critical points is finite, we may make all critical pointspassive after a finite number of perturbations.⊓⊔Example 3.1.10 In the quadratic polynomial family, the bifurcation locus is theboundary of the connectivity locus (or the Mandelbrot set). Indeed, it follows immedialeyfrom the definition 2.3.2 of the Mandelbrot set that its boundary is the activitylocus.Example 3.1.11 The situation is more complicated in the family ( P c,a)(c,a)∈C d−1of degree d poynomials when d ≥ 3 (see subsection 2.1.2). Indeed, Theorem 2.3.5shows that the bifurcation (i.e. activity) locus is not bounded since it coincides withthe boundary of ∪ 0≤i≤d−2 B i (where B i is the set of parameters for which the orbit ofthe critical point c i is bounded) while the connectedness locus ∩ 0≤i≤d−2 B i is bounded.Although we shall not use it, we end this section by quoting a very interestingclassification of the activity situations which is due to Dujardin and Favre (see [DF]Theorem 4).Theorem 3.1.12 Let (f λ ) λ∈Mbe a holomorphic family of degree d rational mapswith a marked critical point c(λ). If c is passive on some connected open subset Uof M then exactly one of the following cases holds:1) c is never preperiodic in U and the closure of its orbits move holomorphicallyon U2) c is persitently preperiodic on U3) the set of parameters for which c is preperiodic is a closed subvariety in U.Moreover, either there exists a persistently attracting cycle attracting c throughoutU, or c lies in the interior of a linearization domain associated to a persistentirrationally neutral periodic point.It is worth emphasize that the proof of that result relies on purely local, andsubtle, arguments.41

3.1.3 Some remarkable parameters• As we already mentionned, the basin of any attracting cycle of a rational mapcontains a critical point. As a consequence, any rational map of degree d has atmost 2d − 2 attracting cycles. Then, using perturbation arguments, Fatou and Juliaproved that the number of non-repelling cycles is bounded by 6d − 6. The sharpbound has been obtained by Shishikura [Sh] using quasiconformal surgery, Epsteinhas given a more algebraic proof based on quadratic differentials (see [Eps2]).Theorem 3.1.13 A rational map of degree d has at most 2d−2 non-repelling cycles.In particular, any degre d rational map cannot have more than 2d − 2 neutralcycles. Shishikura has also shown that the bound 2d−2 is sharp (we will give anotherproof, using bifurcation currents, in subsectiondensShiHyp). Let us mention thatthe Julia set of degree d maps having 2d − 2 Cremer cycles coincides with the fullRiemann sphere. These results motivate the following definition.Definition 3.1.14 The set Shi of degree d Shishikura rational maps is defined byShi = {f ∈ Rat d / f has 2d − 2 neutral cycles}.In a holomorphic family (f λ ) λ∈Mwe shall denote by Shi(M) the set of parameters λfor which f λ is Shishikura.According to Theorem 3.1.9, one has Shi ⊂ Bif. In the last chapter, we willobtain some informations on the geometry of Shi and, in particular, reprove that Shiis not empty.• Any repelling cycle of f ∈ Rat d is an invariant (compact) set on which f isuniformly expanding. Some rational map may be uniformly expanding on muchbigger compact sets. Such compact sets are called hyperbolic and are necessarilycontained in the Julia set. A rational map which is uniformly expanding on its Juliaset is said to be hyperbolic. Let us give a precise definition.Definition 3.1.15 Let f be a rational map. A compact set K of the Riemannsphere is said to be hyperbolic for f if it is invariant (f(K) ⊂ K) and f is uniformlyexpanding on K: there exists C > 0 and M > 1 such that|(f n ) ′ (z)| σ ≥ CM n ; ∀z ∈ K, ∀n ≥ 0.It may happen that a critical orbit is captured by some hyperbolic set and, inparticular, by a repelling cycle. Such rational maps play a very important role inthe study of the parameter space. The reason is that they allow to define transfermaps which carry some informations from the dynamical plane to the parameterspace. A particular attention will be devoted to maps for which all critical orbitsare captured by a hyperbolic set.42

Definition 3.1.16 The set M is of degree d Misiurewicz rational maps is defined byM is = {f ∈ Rat d / all critical orbits of f are captured by a compact hyperbolic set}When the hyperbolic set is an union of repelling cycles the map is said to be stronglyMisiurewicz and the set of such maps is denoted M iss. In a holomorphic family(f λ ) λ∈Mwe shall denote by M is(M) (resp. M iss(M)) the set of parameters λ forwhich f λ is Misiuerewicz (resp. strongly Misiurewicz).Within a holomorphic family (f λ ) λ∈M, one may show that a critical point whoseorbit is captured by a hyperbolic set and leaves this set under a small perturbation isactive. In particular, we have the following inclusion: M is ⊂ Bif. To prove this, onefirst has to construct a holomorphic motion of the hyperbolic set and then linearizealong its orbits (see [G]). When the hyperbolic set is a cycle, the motion is given bythe implicit function theorem and the linearizability is a well known fact.Lemma 3.1.17 Let (f λ ) λ∈Mbe a holomorphic family of degree d rational mapswith a marked critical point c(λ). Assume that f λ has a repelling n-cycle R(λ) :={z λ , f λ (z λ ), · · ·, f n−1λ(z λ )} for λ ∈ U. If fλ k 0(c(λ 0 ) ∈ R(λ 0 ) for some λ 0 ∈ U but notfor all λ ∈ U then c(λ) is active at λ 0 .Proof. We may assume that n = 1 which means that z λ is fixed by f λ . ShrinkingU and linearizing we get a a family of local biholomorphisms φ λ which dependsholomorphically on λ and such that φ λ (0) = z λ and f λ ◦ φ λ (u) = m λ φ λ (u) (see [BM]Théorème II.1 and Remarque II.2). As z λ is repelling, one has |m λ | > 1. Let us setu(λ) := φ −1λ(f λ k p+k(c(λ)), then fλ(c λ ) = f p λ (φ λ(u(λ))) = m λ u(λ) which shows that(c λ ) is not normal at λ 0 since, by assumption, u(λ 0 ) = 0 but u does not vanishidentically on U.⊓⊔f p+kλ• As we already mentionned, a rational map which is uniformly expanding on itsJulia set is called hyperbolic. From a dynamical point of view, the study of suchmaps turns out to be much easier.Definition 3.1.18 The set Hyp of degree d hyperbolic rational maps is defined byHyp = {f ∈ Rat d / f is uniformly expanding on its Julia set}.In a holomorphic family (f λ ) λ∈Mwe shall denote by Hyp(M) the set of parametersλ for which f λ is hyperbolic.There are several characterizations of hyperbolicity. One may show that a rationalmap f is hyperbolic if and only if its postcitical set does not contaminate itsJulia set:f is hyperbolic ⇔ ∪ n≥0 f n (C f ) ∩ J f = ∅.43

As a consequence, in any holomorphic family (f λ ) λ∈M, hyperbolic parametersare stable : Hyp(M) ⊂ S. This characterization also implies that a hyperbolic maphas only attracting or repelling cycles.The Fatou’s conjecture asserts that the hyperbolic parameters are dense in anyholomorphic family, it is an open problem even for quadratic polynomials. Accordingto Mañé-Sad-Sullivan Theorem it can be rephrased as followsFatou’s conjecture 3.1.19 Hyp(M) = S for any holomorphic family (f λ ) λ∈M.let us also mention that Mañé, Sad and Sullivan have shown that hyperbolicand non-hyperbolic parameters cannot coexist in the same stable component (i.econnected component of S).Although the bifurcation locus of the quadratic polynomial family is clearly accumulatedby hyperbolic parameters, this is far from being clear in other familiesand seems to be an interesting question. In the last chapter we will show that parameterswhich, in some sense, produce the strongest bifurcations, are accumulatedby hyperbolic parameters.3.2 Potential theoretic approach3.2.1 Przytycki’s generalized formulaWe want here to establish a fundamental formula which relates the Lyapunov exponent(see definition 1.3.1) and the critical points. The model is a formula due toPrzytycki which deals with the case of polynomials.Theorem 3.2.1 (Przytycki’s formula) Let P be a unitary degree d polynomial,L(P ) its Lyapunov exponent and g P its Green function. ThenL(P ) = ln d + ∑ g P (c)where the sum is taken over the critical points of P counted with multiplicity.Proof. Let us write c 1 , c 2 , · · ·, c d−1 the critical points of P . Then∫L(P ) =C∫ln |P ′ | µ P =C∏d−1ln |d (z − c j )| µ P = ln d +j=1∑d−1∫j=1Cln |z − c j | µ P .Now, since µ P = dd c g P , the formula immediately follows by an integration by parts.⊓⊔44

Remark 3.2.2 As the Green function g P of a polynomial P is positive (see Proposition1.1.12) the above formula implies that L(P ) ≥ ln d.It is more delicate to perform such an integration by part in the case of a rationalmap f. To this purpose we will work in the line bundle O P 1(D) for D := 2(d − 1)which we endow with two metrics, the flat one ‖[z, x]‖ 0 and the Green metric‖[z, x]‖ GF whose potential is the Green function G F of some lift F of f (see subsection1.3.1).In this general situation the integration by part yields the following formula.Proposition 3.2.3 Let f be a rational map of degree d ≥ 2 and F be one of itslifts. Let D := 2(d − 1) and Jac F be the holomorphic section of O P 1(D) induced bydet F ′ . Let g F be the Green function of F on P 1 and µ f be the Green measure of f.ThenL(f) + ln d = ∫ P 1 g F [C f ] − 2(d − 1) ∫ P 1 g F (µ f + ω) + ∫ P 1 ln ‖Jac F ‖ 0 ω.Proof. Let us recall that ‖ · ‖ G = e −Dg F‖ · ‖ 0 . According to Lemma 1.3.4 we have:∫∫∫L(f) + ln d = ln ‖Jac F ‖ GF µ f = ln ‖Jac F ‖ GF dd c g F + ln ‖Jac F ‖ GF ωP 1 P 1 P 1which, after integrating by parts, yields∫∫L(f) + ln d = g F (dd c ln ‖Jac F ‖ GF ) + ln ‖Jac F ‖ GFP 1 P 1ωand by Poincaré-Lelong formula:L(f) + ln d =∫g F ([C f ] − Dµ f ) +P 1 ∫(ln ‖Jac F ‖ 0 − Dg F ) ω.P 1 (3.2.1)When working with a holomorphic family of polynomials (P λ ) M, Przytycki’s formulasays that the Lyapunov function L(P λ ) and the sum of values of the Greenfunction on critical points differ from a constant. In particular, L(P λ ) is a p.s.hfunction on M and these two functions induce the same (1, 1) current on M. Itis then rather clear, using Mañé-Sad-Sullivan Theorem 3.1.9, that this current isexactly supported by the bifurcation locus.We aim to generalize this to holomorphic families of rational maps and willtherefore compute the dd c of the left part of formula 3.2.1. The following formulahas been established in [BB1].45⊓⊔

Theorem 3.2.4 Let (f λ ) Mbe a holomorphic family of degree d rational maps whichadmits a holomorphic family of lifts (F λ ) M. Let p M (resp p P 1) be the canonicalprojection from M × P 1 onto M (resp.P 1 ). Thendd c L(λ) = (p M ) ⋆((ddcλ,z g λ (z) + ˆω) ∧ [C] ) (3.2.2)where C := {(λ, z) ∈ M × P 1 / z ∈ C λ }, g λ is the Green function F λ on P 1 , L(λ)the Lyapunov exponent of f λ and ˆω := p ⋆ P 1 ω.Moreover, the function λ ↦→ ∫ P 1 g λ (µ λ + ω) is pluriharmonic on M.Proof. We may work locally and define a holomorphic section Jac λ of O P 1(D) inducedby det F ′ λ . Then ˜Jac(λ, [z]) := (λ, Jac λ ([z])) is a holomorphic section of theline bundle M × O P 1(D) over M × P 1 .Let us rewrite Proposition 3.2.3 on the form L(λ) + ln d = H(λ) − D B(λ) whereH(λ) := ∫ P 1 g λ [C λ ] + ∫ P 1 ln ‖Jac λ ‖ 0 ω, B(λ) := ∫ P 1 g λ (µ λ + ω) and D := 2(d − 1).We first compute dd c H. Let Φ denote a (m − 1, m − 1) test form where m isthe dimension of M. Then 〈dd c H, Φ〉 = I 1 + I 2 where I 1 := ∫ M ddc Φ ∫ P 1 g λ [C λ ] andI 2 := ∫ M ddc Φ ∫ P 1 ln ‖Jac λ ‖ 0 ω.Slicing and then integrating by parts, we get∫ ∫∫I 1 = dd c Φ (g[C]) λ= (p M ) ⋆ (dd c Φ) ∧ (g[C]) =M P 1 M×P∫1= (p M ) ⋆ Φ ∧ dd c g ∧ [C].M×P 1Similarly and then using Poincaré-Lelong identity dd c ln ‖˜Jac‖ 0 = [C] − Dˆω onegets∫ ( ) ∫I 2 = dd c Φ ln ‖˜Jac‖ 0 ˆω = (p M )M∫P ⋆ (dd c Φ) ∧ ln ‖˜Jac‖ 0 ˆω =1 λ M×P∫1 ∫= (p M ) ⋆ Φ ∧ dd c ln ‖˜Jac‖ 0 ∧ ˆω = (p M ) ⋆ Φ ∧ ˆω ∧ [C].M×P 1 M×P 1This shows that dd c H = (p M ) ⋆((ddcλ,z g λ (z) + ˆω) ∧ [C] ) .It remains to show that dd c B = 0. It seems very difficult to prove this bycalculus, we will use a trick which exploits the dynamical situation. If one replacethe family (f λ ) Mby the family (fλ 2) M then L becomes 2L and ddc H becomes2dd c H while B is unchanged. Thus, applying Proposition 3.2.3 to (f λ ) Mand takingdd c yields dd c L = dd c H − 2(d − 1)dd c B but, with the family (fλ 2) M, this yields2dd c L = 2dd c H − 2(d 2 − 1)dd c B. By comparison one obtains dd c B = 0. ⊓⊔46

Remark 3.2.5 In [BB1] the formula 3.2.2 has been generalized to the case of holomorphicfamilies of endomorphisms of P k . It becomesdd c L(λ) = (p M ) ⋆((ddcλ,z g λ (z) + ˆω) k ∧ [C] )where L(λ) is now the sum of Lyapunov exponents of f λ with respect to the Greenmeasure µ λ .3.2.2 Lyapunov exponent and bifurcation currentWe have seen with Mañé-Sad-Sullivan Theorem 3.1.9 that the bifurcations, withina holomorphic family of degree d rational maps (f λ ) λ∈M, are due to the activity ofthe critical points (see in particular Lemma 3.1.7). We will use this and the formulagiven by Theorem 3.2.4, to define a closed positive (1, 1)-current T bif on M whosesupport is the bifurcation locus and which admits the Lyapunov function as globalpotential. Besides, we will also introduce a collection of 2d − 2 closed positive (1, 1)-currents which detect the activity of each critical point and see that the bifurcationcurrent T bif is the sum of these currents. Here are the formal definitions.Definition 3.2.6 Let (f λ ) λ∈Mbe any holomorphic family of degree d rational maps.with marked critical points {c i (λ); 1 ≤ i ≤ 2d − 2}. The activity current T i of themarked critical point c i is defined byT i := (p M ) ⋆((ddcλ,z g λ (z) + ˆω) ∧ [C i ] ) .where C i is the graph {(λ, c i (λ); λ ∈ M} in M × P 1 and g λ is the Green functionon P 1 of F λ for some (local) holomorphic family of lifts (F λ ).The bifurcation current T bif is defined byT bif := dd c Lwhere L is the Lyapunov exponent of the family (f λ ) λ∈M.We start by giving local potentials for the activity currents.Lemma 3.2.7 Let F λ be a local holomorphic family of lifts of f λ and G λ be theGreen function of F λ . Let ĉ i (λ) be a local lift of c i (λ). Then G λ (ĉ i (λ)) is a localpotential of T i .Proof. This is a straightforward computation using the fact that, for any local sectionσ of the canonical projection π : C 2 \ {0} → P 1 , the function G λ (σ(z)) is alocal potential of dd c λ,z g λ(z) + ˆω (see Proposition 1.1.6).⊓⊔The following result has been originally proved by DeMarco in [DeM1], [DeM2].47

Theorem 3.2.8 Let (f λ ) λ∈Mbe any holomorphic family of degree d rational mapswith marked critical points {c i (λ); 1 ≤ i ≤ 2d − 2}. The support of the activitycurrent T i is the activity locus of the marked critical point c i .The support of the bifurcation current T bif is the bifurcation locus of the family(f λ ) λ∈Mand T bif = ∑ 2d−21T i .Proof. Let us first show that c i is passive on the complement of Supp T i . If T i = 0 ona small ball B ⊂ M then, by Lemma 3.2.7, G λ (ĉ i (λ)) is pluriharmonic and thereforeequal to ln |h i (λ)| for some non-vanishing holomorphic function h i on B. Replacingĉ i (λ) by ĉi(λ) one gets, thanks to the homogeneity property of G h i (λ) λ (see Proposition1.1.6), G λ (ĉ i (λ)) = 0. This implies that{F n λ (ĉ i(λ)) / n ≥ 1, λ ∈ B} ⊂ ∪ λ∈B G −1λ {0}which, after reducing B, is a relatively compact subset of C 2 . Montel’s theoremthen tells us that (F n λ (ĉ i(λ))) nand thus (f n λ (c i(λ))) nare normal on B.Let us now show that T i vanishes where c i is passive. Assume that a subsequence(f n kλ(c i(λ))) kis uniformly converging on a small ball B ⊂ M. Then we may find alocal section σ of π : C 2 \ {0} → P 1 such that F n kλ(ĉ i(λ)) = h nk (λ) · σ ◦ f n kλ(c i(λ))where h nk is a non-vanishing holomorphic function on B. As G λ ◦ F λ = dG λ (seeProposition 1.1.6), this yieldsG λ (ĉ i (λ)) = d k( −n ln |hnk (λ)| + G λ ◦ σ ◦ f n kλ (c i(λ)) )which, after taking dd c and making k → +∞, implies that T i vanishes on B.That dd c L = T bif = ∑ 2d−21T i follows immediately from Theorem 3.2.4. Then,Mañé-Sad-Sullivan Theorem 3.1.9 implies that the support of T bif is the bifurcationlocus.⊓⊔It is important to stress here that the identity dd c L = T bif = ∑ 2d−21T i maybe seen as a potential-theoretic expression of Mañé-Sad-Sullivan theory. More concretely,the expression T bif = dd c L will be used to investigate the set of parametersShi while the expression T bif = ∑ 2d−21T i will be used for the parameters M is (seesubsection 3.1.3).In the next chapters, we will deeply use the fact that the Lyapunov function is apotential of T bif , together with the approximation formula 1.3.7, to analyse how thehypersurfaces P er n (w) may shape the bifurcation locus.The continuity of the Lyapunov function will turn out to be extremely useful for thisstudy. Mañé was the first to establish the continuity of the Lyapunov function L(see [Mañ]) , using Theorem 3.2.4 and Lemma 3.2.7 one may see that this functionis actually Hölder continuous.Theorem 3.2.9 The Lyapunov function of any holomorphic family of degree d rationalmaps is p.s.h and Hölder continuous.48

Proof. According to Theorem 3.2.4 and lemma 3.2.7 the functions L and ∑ G λ (ĉ i (λ))differ from a pluriharmonic function. The conclusion follows from the fact that G λ (z)is Hölder continuous in (λ, z) (see [BB1] Proposition 1.2.).⊓⊔3.2.3 DeMarco’s formulaUsing Theorem 3.2.4 we will get an explicit version of the formula given by Proposition3.2.3. This result was first obtained by DeMarco who used a completelydifferent method. We refer to the paper of Okuyama [O] for yet another proof.The key will be to compute the integral ∫ gP 1 F (µ f + ω) which appears in theformula given by Proposition 3.2.3. To this purpose we shall use the resultant of ahomogeneous polynomial map F of degree d on C 2 . The space of such maps canbe identified with C 2d+2 . The resultant Res F of F polynomialy depends on Fand vanishes if and only if F is degenerate. Moreover Res (z1 d, zd 2 ) = 1 and Res is2d-homogeneous: Res aF = a 2d Res F .Lemma 3.2.10 ∫ P 1 g F (µ f + ω) = 1d(d−1) ln |Res F | − 1 2Proof. The function B(F ) := ∫ P 1 g F (µ f + ω) is well defined on C 2d+2 \ Σ where Σ isthe hypersurface where Res vanishes. Moreover, according to Theorem 3.2.4, B ispluriharmonic. As B is locally bounded from above, it extends to some p.s.h functionthrough Σ. Then, by Siu’s theorem, there exists some positive constant c such thatdd c B(F ) = c dd c ln |ResF | which means that B − c ln |ResF | is pluriharmonic onC 2d+2 .Let ϕ be a non-vanishing holomorphic function on C 2d+2 such thatB − c ln |ResF | = ln |ϕ|.Using the homogeneity of Res and the fact that B(aF ) = 2 ln |a| + B(F ) (oned−1easily checks that g aF = 1 ln |a| + g d−1 F ) one gets:|ϕ(aF )| = |a| 2d−1 −2cdMaking a → 0 one sees that c = 1 and ϕ is constant. To compute this constantd(d−1)one essentially tests the formula on F 0 := (z1 d, zd 2 ) (see [BB1] Proposition 4.10). ⊓⊔We are now ready to prove the main result of this subsection.Theorem 3.2.11 Let f be a rational map of degree d ≥ 2 and F be one of itslifts. Let G F be the Green function of F on C 2 and Res F the resultant of F . Ifĉ 1 , ĉ 2 , · · ·, ĉ 2d−2 are chosen so that detF ′ (z) = ∏ 2d−2j=1 ĉj ∧ z then49

2d−2∑L(f) + ln d = G F (ĉ j ) − 2 ln |Res F |.dj=1Proof. Taking Lemma 3.2.10 into account, the formula given by Proposition 3.2.3becomesL(f) + ln d = ∫ ()1gP 1 F [C f ] − 2(d − 1) ln |Res F | − 1 + ∫ ln ‖Jacd(d−1) 2 P 1 F ‖ 0 ω.Observe that ω = π ⋆ m where m := ( dd c ln + ‖ · ‖ ) 2is the normalized Lebesguemeasure on the euclidean unit sphere of C 2 . Then∫ln |detF ′ | m = ∫ ln ( e −D‖·‖ |detF ′ | ) m = ∫ ln ‖JC 2 C 2 C 2 F ◦π‖ 0 m = ∫ ln ‖JacP 1 F ‖ 0 ω.Let us pick U j in the unitary group of C 2 such that U −1j (ĉ j ) = (‖ĉ j ‖, 0). ThenU j (z) ∧ ĉ j = −z 2 ‖ĉ j ‖ and, since ∫ ln |zC 2 2 | m = − 1 one gets2∫ln ‖JacP 1 F ‖ 0 ω = ∫ ln |detF ′ | m = ∑ ∫C 2 jln |UC 2 j (z) ∧ ĉ j | = ∑ j ‖ĉ j‖ − (d − 1).On the other hand, ∫ gP 1 F [C f ] = ∑ j g F ◦ π(ĉ i ) = ∑ j G F (ĉ j ) − ∑ j ln ‖ĉ j‖ and theconclusion follows.⊓⊔50

Chapter 4Equidistribution towards thebifurcation current4.1 Distribution of critically periodic parametersThe aim of this section is to present a result due to Dujardin and Favre (see [DF])about the asymptotic distribution of degree d polynomials which have a pre-periodiccritical point. We will work in the context of polynomial families, this will allow usto modify the original proof and significantly simplify it. We refer to the paper ofDujardin-Favre for a proof working in any holomorphic family of rational maps.4.1.1 Statement and a general strategyWe work here in the family ( P c,a)(c,a)∈C d−1 of degree d polynomials which has beenintroduced in the subsection 2.1.2. Let us recall that P c,a is the polynomial of degreed whose critical points are (0 = c 0 , c 1 , · · ·, c d−2 ) and such that P c,a (0) = a d .For 0 ≤ i ≤ d − 2 and 0 ≤ k < n, we denote by Per(i, n, k) the hypersurface ofC d−1 defined byPer(i, n, k) := {(c, a) ∈ C d−1 / P n c,a (c i) = P k c,a (c i)}.The results we want to establish is the following; it has been first proved byDujardin and Favre in [DF]Theorem 4.1.1 In the family of degree d polynomials, for any sequence of integers(k n ) n such that 0 ≤ k n < n one has ∑ d−2i=0 lim n d −n [Per(i, n, k n )] = T bif .Let us now explain the principle of the proof and fix a few notations. To simplify,we shall write λ the parameters (c, a) ∈ C d−1 .51

The bifurcation current T bif is given by T bif = ∑ d−2i=0 ddc g λ (c i ) (see Lemma 3.2.7)where g λ is the Green function of P λ (see the subsection 2.3.1). It thus suffices toshow that for any fixed 0 ≤ i ≤ d − 2 the following sequence of potentialsh n (λ) := d −n ln |Pλ n(c i) − P knλ(c i)|converges in L 1 loc to g λ(c i ).To this purpose we shall compare these potentials with the functionsg n (λ) := d −n ln max ( 1, |P n λ (c i)| )which do converge locally uniformly to g λ (c i ).The first important point is to check that the sequence (h n ) n is locally uniformlybounded from above. We shall actually prove a little bit more.Lemma 4.1.2 For any compact K ⊂ C d−1 and any ɛ > 0 there exists an integern 0 such that h n | K ≤ g n | K + ɛ for n ≥ n 0 .Proof. It is not difficult to see that there exists R ≥ 1 such that(1 − ɛ)|z| dn ≤ |P n λ (z)| ≤ (1 + ɛ)|z| dn (4.1.1)for every λ ∈ K, every n ∈ N and every |z| ≥ R.We now proceed by contradiction and assume that there exists λ pn p → +∞ such that h np (λ p ) ≥ g np (λ p ) + ɛ. This means that∈ K and|P npλ p(c i ) − P knpλ p(c i )| ≥ e ɛdnp max ( 1, |P npλ p(c i )| ) . (4.1.2)Let us set B p := P knpλ p(c i ). By 4.1.2 we have lim p |B p | = +∞ and thus |B p | ≥ R forp big enough. Then, using 4.1.1, one may writeλ p(c i ) = P np−knp(B p ) = (u p B p ) dnp−knpP npwhere (1 − ɛ) ≤ |u p | ≤ (1 + ɛ) and the estimate 4.1.2 becomesλ p|(u p B p ) dnp−knp − B p | ≥ e ɛdnp |u p B p | dnp−knp .This is clearly impossible when p → +∞.⊓⊔To prove that (h n ) n converges in L 1 loc to g λ(c i ) we shall use a well known compactnessprinciple for subharmonic functions:Theorem 4.1.3 Let (ϕ j ) be a sequence of subharmonic functions which is locallyuniformly bounded from above on some domain Ω ⊂ R n . If (ϕ j ) does not convergeto −∞ then a subsequence (ϕ jk ) converges in L 1 loc (Ω) to some subharmonic functionϕ. In particular, (ϕ j ) converges in L 1 loc (Ω) to some subharmonic function ϕ if itconverges pointwise to ϕ.52

We will also need to following classical resultLemma 4.1.4 (Hartogs) Let (ϕ j ) be a sequence of subharmonic functions and g bea continuous function defined on some domain Ω ⊂ R n . If lim sup j ϕ j (x) ≤ g(x) forevery x ∈ Ω then, for any compact K ⊂ Ω and every ɛ > 0 one has ϕ j (x) ≤ g(x) + ɛon K for j big enough.By Lemma 4.1.2 the sequence (h n ) n is locally uniformly bounded from aboveand therefore, according to the Theorem 4.1.3, we have to check that (h n ) n does notconverge to −∞ and that g λ (c i ) is the only limit value of (h n ) n for the L 1 loc convergence.To prove this unicity statement we will combine the following maximum-typeprinciple with our knowledge of the behaviour of the bifurcation locus at infinity inpolynomials families (see Theorem 2.3.5).Lemma 4.1.5 Let ϕ, ψ be two p.s.h functions on C k . Assume that:i) ψ is continuousii) ϕ ≤ ψiii) Supp (dd c ϕ) ⊂ Supp (dd c ψ)iv) ϕ = ψ on Supp (dd c ψ)v) for any λ 0 ∈ C k there exists a complex line L through λ 0 such that ϕ = ψ onthe unbounded component of L \ ( L ∩ Supp (dd c ψ) ) .Then ϕ = ψ.Proof. Because of iv), we only have to show that ϕ(λ 0 ) = ψ(λ 0 ) when λ 0 lies in thecomplement of Supp (dd c ψ). According to (v), we may find a complex line L in C kcontaining λ 0 and such that ψ and ϕ coincide on the unbounded component Ω ∞ ofL \ ( L ∩ Supp (dd c ψ) ) . We may therefore assume that λ 0 /∈ Ω ∞ . By i), iii) and iv)ϕ| L coincides with the continuous function ψ| L on Supp ∆ϕ| L which, by the continuityprinciple, implies that ϕ| L is continuous. Let Ω 0 be the (bounded) componentof L \ ( L ∩ Supp (dd c ψ) ) containing λ 0 . The continuous function ( ϕ − ψ ) | L vanisheson bΩ 0 (see iv)), is harmonic on Ω 0 (see iii) and iv)) and negative (see ii)). Themaximum principle now implies that ϕ(λ 0 ) = ψ(λ 0 ).⊓⊔The following technical lemma will play an important role in the proof of Theorem4.1.1. It shows in particular that (h n ) n doest not converge to −∞.Lemma 4.1.6 If c i belongs to some attracting bassin of P λ0 then there exists aneighbourhood V 0 of λ 0 such that sup n sup V0(hn − g n)≥ 0.53

Proof. If V 0 is a sufficently small neighbourhood of λ 0 then P n λ (c i) → a λ where a λ isan attracting cycle of P λ for every λ ∈ V 0 . We will assume that P λ (a λ ) = a λ .Let us now proceed by contradiction and assume that there exists ɛ > 0 such that|P n λ (c i ) − P knλ (c i)| ≤ e −ɛdn max ( 1, |P n λ (c i )| ) , ∀λ ∈ V 0 , ∀n. (4.1.3)Since P n λ (c i) → a λ , 4.1.3 would imply|P n λ (c i ) − P knλ (c i)| ≤ Ce −ɛdn , ∀λ ∈ V 0 , ∀n. (4.1.4)The estimate 4.1.4 implies that a λ is a super-attracting fixed point for any λ ∈ V 0which, in turn, implies that a λ = ∞ for all λ ∈ V 0 . But in that case we would have|P knλ(c i)| ≤ 1|P n 2 λ (c i)| for n big enough and the estimate 4.1.3 would be violated. ⊓⊔4.1.2 Proof of Theorem 4.1.1We use here the notations introduced at the beginning of the previous subsection.According to Theorem 4.1.3 we have to show that g := g λ (c i ) = lim g n is the onlylimit value of the sequence (h n ) n for the L 1 loc convergence (by Lemmas 4.1.2 and4.1.6, this sequence of p.s.h functions is locally uniformly bounded and does notconverge to −∞). Assume that (after taking a subsequence!) h n is converging inL 1 loc to h. To prove that the functions h and g coincide we shall check that theysatisfy the assumptions of Lemma 4.1.5.Our modification of the original proof essentially stays in the third step.First step: h ≤ g.Let B 0 be a ball of radius r centered at λ 0 and let ɛ > 0. By the mean valueproperty we have∫∫h(λ 0 ) ≤ 11|B 0 | B 0h = lim n |B 0 | B 0h nbut, according to Lemma 4.1.2, h n ≤ g n + ɛ on B 0 for n big enough and thus∫∫1h(λ 0 ) ≤ ɛ + lim n |B 0 | B 0g n = ɛ + 1|B 0 | B 0g.As g is continuous, the conclusion follows by making r → 0 and then ɛ → 0.Second step: h = g on Supp dd c g.Combining Lemma 4.1.6 and the result of first step we will first see that (h − g)vanishes when c i is captured by an attracting basin.Suppose to the contrary that c i is captured by an attracting cycle of P λ0 and54

(h − g)(λ 0 ) < 0. As the function (h − g) is upper semi-continuous, we may shrinkV 0 so that (h − g) ≤ −ɛ < 0 on V 0 . Now, as c i is passive on V 0 , the function(h − g) is p.s.h on V 0 and, after shrinking V 0 again, Hartogs Lemma 4.1.4 impliesthat (h n − g n ) ≤ − ɛ on V 2 0 for n big enough. This contradicts Lemma 4.1.6.Now, if λ 0 ∈ Supp dd c g then λ 0 = lim k λ k where λ k is a parameter for which c iis captured by some attracting cycle (see Lemma 3.1.8). As (h − g) is uper semicontinuouswe get (h − g)(λ 0 ) = 0.Third step: Supp dd c h ⊂ Supp dd c g.Let Ω be a connected component of C d−1 \ Supp dd c g. We have to show that his pluriharmonic on Ω. We proceed by contradiction. If dd c h does not vanish on Ωthen there exists some n 0 for which the hypersurfaceH := {P n 0λ (c i) − P kn 0λ(c i ) = 0}meets Ω. When λ ∈ H then c i is captured by a cycle since P kn 0λ(c i ) =: z(λ) satisfiesP m 0λ (z(λ)) = P m 0λ◦ P kn 0λ(c i ) = P kn 0λ(c i ) = z(λ) for m 0 := n 0 − k n0 > 0.Let us show that z(λ) is a neutral periodic point. We first observe that thevanishing of dd c g on Ω forces z(λ) to be non-repelling and thus | ( )P m 0 ′ ( )λ z(λ) | ≤ 1on H ∩ Ω.Let us now see why z(λ) cannot be attracting. If this would be the case then,according to Lemma 4.1.6, we would have h(λ 0 ) = g(λ 0 ) for a certain λ 0 ∈ H ∩ Ω.As (h − g) is negative and p.s.h on Ω this implies, via the maximum principle, thath = g on Ω. This is imposible since dd c h is supposed to be non vanishing on Ω.We thus have ( )P m 0 ′ ( )λ z(λ) = eiν 0on H ∩ Ω and therefore z(λ) belongs to a neutralcycle whose period divides m 0 and whose multiplier is a root of e iν 0.In other words, H ∩ Ω is contained in a finite union of hypersurfaces of the formPer n (e iθ ). This implies thatH ⊂ Per n0 (e iθ 0).for some integer n 0 and some real number θ 0 .Finally, using a global argument, we will see that this is impossible. Let us recallthe following dynamical fact.Lemma 4.1.7 Every polynomial which has a neutral cycle also has a bounded, nonpreperiodic,critical orbit.Thus, when λ ∈ H, the polynomial P λ has two distinct bounded critical orbits;the orbit of c i which is preperiodic and the orbit of some other critical point55

which is given by the above lemma. This shows that H cannot meet the line{c 0 = c 1 = · · · = c d−2 = 0} := M d since the corresponding polynomials (whichare given by 1 d zd + a d a ∈ C) have only one critical orbit. We will now work inthe projective compactification of C d−1 introduced in subsection 2.1.2. By Theorem2.3.5, H and M d cannot meet at infinity. This contradicts Bezout’s theorem.Fourth step: for any λ 0 ∈ C d−1 there exists a complex line L through λ 0 suchthat h = g on the unbounded component of L \ ( L ∩ Supp(dd c g) ) .Here one uses again Theorem 2.3.5 to pick a line L through λ 0 which meetsinfinity at some point ξ 0 /∈ Supp dd c g. Then, for any λ in the unbounded componentof L \ ( L ∩ Supp (dd c g) ) the critical point c i belongs to the super-attracting basinof ∞ and thus, as we saw in second step, h(λ) = g(λ).⊓⊔.4.2 Distribution of rational maps with cycles of agiven multiplierLet f : M × P 1 → P 1 be an arbitrary holomorphic family of degree d ≥ 2 rationalmaps.We want to investigate the asymptotic distribution of the hypersurfaces P er n (w)in M when |w| < 1. Concretely, we will consider the current of integration [Per n (w)]or, more precisely, the currents[Per n (w)] := dd c ln |p n (λ, w)|where p n (·, w) are the canonical defining functions for the hypersurfaces Per n (w)given by Theorem 2.2.1. We ask if the following convergence occurs:lim n1d n [Per n (w)] = T bif .The question is easy to handle when |w| < 1, more delicate when |w| = 1 and widelyopen when |w| > 1.4.2.1 The case of attracting cyclesWe aim to prove the following general result result (see [BB2])Theorem 4.2.1 For any holomorphic family of degree d rational maps ( f λ)λ∈M onehas d −n [Per n (w)] → T bif when |w| < 1.Proof. Let us setL n (λ, w) := d −n ln |p n (λ, w)|.56

Since, by definition, T bif = dd c L(λ) where L(λ) be the Lyapunov exponent of(P 1 , f λ , µ λ ) and µ λ is the Green measure of f λ , all we have to show is that L nconverges to L in L 1 loc (M). Here again we shall use the compactness principle forsubharmonic functions (see Theorem 4.1.3).The situation is purely local and therefore, taking charts, we may assume thatM = C k . We write the polynomials p n as follows :p n (λ, w) =:N d (n)∏i=1(w − wn,j (λ) ) .Using use the fact that d −n N d (n) ∼ 1 (see Theorem 2.2.1) one sees that the sequencesL n is locally uniformly bounded fromnabove.According to Theorem 2.2.1, the set {w n,j (λ) / w n,j (λ) ≠ 1} coincides with theset of multipliers of cycles of exact period n (counted with multiplicity) from whichthe cycles of multiplier 1 are deleted. We thus haveN d (n)∑j=1ln + |w n,j (λ)| = 1 n∑p∈R ∗ n(λ)ln |(f n ) ′ (p)| (4.2.1)where R ∗ n(λ) := {p ∈ P 1 / p has exact period n and |(f n λ )′ (p)| > 1}. Since f λhas a finite number of non-repelling cycles (Fatou’ theorem), one sees that thereexists n(λ) ∈ N such thatn ≥ n(λ) ⇒ |w n,j (λ)| > 1, for any 1 ≤ j ≤ N d (n). (4.2.2)By 4.2.1 and 4.2.2, one getsN d (n)∑L n (λ, 0) = d −nj=1N d (n)∑ln |w n,j (λ)| = d −nj=1for n ≥ n(λ) which, by Theorem 1.3.7, yields:ln + |w n,j (λ)| = d−nn∑R ∗ n (λ) ln |(f n ) ′ (p)|limnL n (λ, 0) = L(λ), ∀λ ∈ M. (4.2.3)If now |w| < 1, it follows from 4.2.2 that L n (λ, w)−L n (λ, 0) = d ∑ −n j ln |w n,j(λ)−w||w n,j (λ)|and ln(1 − |w|) ≤ ln |w n,j(λ)−w||w n,j≤ ln(1 + |w|) for 1 ≤ j ≤ N(λ)| d (n) and n ≥ n(λ) . Wethus getd −n N d (n) ln(1 − |w|) ≤ |L n (λ, w) − L n (λ, 0)| ≤ d −n N d (n) ln(1 + |w|)57

for n ≥ n(λ). Using 4.2.3 and the fact that d −n N d (n) ∼ 1 we obtain lim n n L n (λ, w) =L(λ) for any (λ, w) ∈ M × ∆. The L 1 loc convergence of L n(·, w) now follows immediatelyfrom Theorem 4.1.3.⊓⊔Remark 4.2.2 We have proved that L n (λ, w) := d −n ln |p n (λ, w)| converges pointwiseto L(λ) on M when |w| < 1.The above discussion shows that the pointwise convergence of L n (λ, w) to L(and therefore the convergence d −n [Per n (w)] → T bif ) is quite a straightforward consequenceof Theorem 1.3.7 when |w| < 1. However, when |w| ≥ 1 and λ is anon-hyperbolic parameter, the control of L n (λ, w) = d ∑ −n ln |w − w n,j (λ)| is verydelicate because f λ may have many cycles whose multipliers are close to w. Thisis why we introduce the p.s.h functions L + n which both coincide with L n on thehyperbolic components and are quite easily seen to converge nicely. These functionswill be extremely helpfull later.Definition 4.2.3 Let p n (λ, w) =: ∏ N d (n)j=1(w−w n,j (λ)) be the polynomials associatedto the family f by Theorem 2.2.1. The p.s.h functions L + n are defined by:N d (n)∑L + n (λ, w) := d −nj=1ln + |w − w n,j (λ)|.The interest of considering these functions stays in the next lemma.Lemma 4.2.4 The sequence L + n converges pointwise and in L 1 loc to L on M × C.For every w ∈ C the sequence L + n (·, w) converges in L1 loc to L on M.Proof. We will show that L + n (·, w) converges pointwise to L on M for every w ∈ C.As (L + n ) n is locally uniformly bounded, this implies the convergence of L + n (·, w) inL 1 loc (M) (Theorem 4.1.3) and the convergence of L+ n in L 1 loc (M × C) then follows byLebesgue’s theorem.As L n (λ, 0) → L(λ) (see Remark 4.2.2), we have to estimate L + n (λ, w)−L n (λ, 0) =:ɛ n (λ, w) on M. Let us fix λ ∈ M, w ∈ C and pick R > |w|. Since f λ has a finitenumber of non-repelling cycles (Fatou’ theorem), one sees that there exists n(λ) ∈ Nsuch thatn ≥ n(λ) ⇒ |w n,j (λ)| > 1, for any 1 ≤ j ≤ N d (n).We may then decompose ɛ n (λ, w) in the following way:58

ɛ n (λ, w) = d −n∑1≤|w n,j (λ)||w| + 1, one has:ln(1 −RR + 1 ) ≤ ln |w n,j(λ)| − R≤ ln |w n,j(λ) − w||w n,j (λ)| |w n,j (λ)|≤ ln |w n,j(λ)| + R≤ ln(1 +R|w n,j (λ)|R + 1 ).As the functions L + n and L n coincide on hyperbolic components, the above lemmawould easily yield the convergence of d −n [Per n (w)] towards T bif for any w ∈ C if thedensity of hyperbolic parameters in M was known. The remaining of this section is,in some sense, devoted to overcome this difficulty. We shall first do this in a generalsetting by averaging the multipiers. Then we will restrict ourself to polynomialfamilies and, using the nice distribution of hyperbolic parameters near infinity, willshow that d −n [Per n (e iθ )] converges towards T bif .⊓⊔4.2.2 Averaging the multipliersAlthough the convergence of lim n1d n [Per n (w)] to T bif is not clear when |w| ≥ 1, oneeasily obtains the convergence by averaging over the argument of the multiplier w.The following result is due to Bassanelli and Berteloot [BB2].Theorem 4.2.5 For any holomorphic family of degree d rational maps ( f λ∫)λ∈M onehas d−n 2π[Per 2π 0 n(re iθ )] dθ → T bif , when r ≥ 0.Proof.One essentially has to investigate the following sequences of p.s.h functions∫ 2πL r d−nn (λ) := ln |p n (λ, re iθ )| dθ.2π 059

We will see that L r ln rn (λ) ≥ Cnconverges to L in L 1 loc (M).where C only depends on the family and that Lr nFor that, we essentially will compare L r n with L n(λ, 0) = L 0 n by using the formula∫ln max(|a|, r) = 1 2πln |a − re iθ |dθ. Indeed, writting2π 0this formula yieldsp n (λ, w) =:N d (n)∏i=1(w − wn,j (λ) )L r n (λ) = 12πd n ∫ 2π0ln ∏ jd −n ∑ j|re iθ − w n,j (λ)|dθ = (4.2.4)ln max(|w n,j (λ)|, r). (4.2.5)According to Theorem 2.2.1, the set {w n,j (λ) / w n,j (λ) ≠ 1} coincides with theset of multipliers of cycles of exact period n (counted with multiplicity) from whichthe cycles of multiplier 1 are deleted. Using use the fact that d −n N d (n) ∼ 1 (see nTheorem 2.2.1) one sees that the sequence L r n (λ) is locally bounded from aboveand is uniformly bounded from below by C ln r . Since f n λ has a finite number ofnon-repelling cycles (Fatou’ theorem), one sees that there exists n(λ) ∈ N such thatn ≥ n(λ) ⇒ |w n,j (λ)| > 1, for any 1 ≤ j ≤ N d (n).Now we deduce from 4.2.4 that for n ≥ n(λ):L r n(λ) = d −n ∑ jln |w n,j (λ)| + d −nL n (λ, 0) + d −n∑1≤|w n,j (λ)|

dd c L r n = ∫ d−n 2π[P er 2π 0 n(re iθ )]dθ.For this purpose it suffices to check that ln |p n (λ, re iθ )| is locally integrable.Let K be a compact subset of M and c n be an upper bound for ln |p n (λ, re iθ )| onK × [0, 2π]. Then, the negative function ln |p n (λ, e iθ )| − c n is indeed integrable onK × [0, 2π] as it follows from the fact that L r ln rn (λ) ≥ C : n∫( ∫ 2π∫∫( ) )ln |pn (λ, re iθ )| − c n dθ dV = 2πdnL r n dV − 2πc n dVK 0KK(≥ d n C ln r ) ∫n− c n 2π dV.K⊓⊔ln |p n (λ, re iθ )| dθ is point-Remark 4.2.6 We have proved that L r d−nn (λ) :=2πwise converging to L(λ) on M.∫ 2π0The following result is essentially a potential-theoretic consequence of the formerone. It implicitely contains some information about the convergence of d −n [Per n (w)]for arbitrary choices of w but seems hard to improve without using further dynamicalproperties (see [BB3]).Theorem 4.2.7 For family of degree d rational maps ( f λ)λ∈Mone hasd −n dd c (λ,w) ln |p n(λ, w)| → dd c L(λ)where p n (·, w) are the canonical defining functions for the hypersurfaces Per n (w)given by Theorem 2.2.1.Proof. Let us setL n (λ, w) := d −n ln |p n (λ, w)|.As we have seen in the two last subsections (see remarks 4.2.2 and 4.2.6)L r n(λ) := d−n2π∫L n (λ, 0) → L(λ)2πln |p0 n (λ, re iθ )| dθ → L(λ) for any r ≥ 0.Let us also recall that the function L is continuous on M (see Theorem 3.2.9).As the functions L n are p.s.h and the sequence (L n ) n is locally uniformly boundedfrom above, we shall again use the compacity properties of p.s.h functions givenby Theorem 4.1.3. Since L n (λ, 0) converges to L(λ), the sequence (L n ) n does not61

converge to −∞ and it therefore suffices to show that, among p.s.h functions onM × C, the function L is the only possible limit for (L n ) n in L 1 loc (M × C).Let ϕ be a p.s.h function on M × C and (L nj ) j a subsequence of (L n ) n whichconverges to ϕ in L 1 loc (M × C). Pick (λ 0, w 0 ) ∈ M × C. We have to prove thatϕ(λ 0 , w 0 ) = L(λ 0 ).Let us first observe that ϕ(λ 0 , w 0 ) ≤ L(λ 0 ). Take a ball B ɛ of radius ɛ and centeredat (λ 0 , w 0 ) ∈ M ×C. By the submean value property and the L 1 loc - convergenceof L + n (see Lemma 4.2.4) we have:ϕ(λ 0 , w 0 ) ≤ 1 ∫∫1ϕ dm = lim L nj dm|B ɛ | B j ɛ|B ɛ | B∫ɛ1≤ lim L + n j |B ɛ |jdm = 1 ∫L dmB ɛ|B ɛ | B ɛmaking then ɛ → 0, one obtains ϕ(λ 0 , w 0 ) ≤ L(λ 0 ) since L is continuous.Let us now check that lim sup j L nj (λ 0 , w 0 e iθ ) = L(λ 0 ) for almost all θ ∈ [0, 2π]. Letr 0 := |w 0 |. By Lemma 4.2.4 L + n converges pointwise to L and therefore:lim supjL nj (λ 0 , w 0 e iθ ) ≤ lim sup L + n j(λ 0 , w 0 e iθ ) = L(λ 0 ).jOn the other hand, by pointwise convergence of L r 0nhave:L(λ 0 ) = limnL r 0n (λ 0) = lim supj12π12π∫ 2π0∫ 2π0to L and Fatou’s lemma weL nj (λ 0 , r 0 e iθ )dθ ≤lim sup L nj (λ 0 , r 0 e iθ )dθjand the desired property follows immediately.To end the proof we argue by contradiction and assume that ϕ(λ 0 , w 0 ) < L(λ 0 ). Asϕ is upper semi-continuous and L continuous, there exists a neighbourhood V 0 of(λ 0 , w 0 ) and ɛ > 0 such thatϕ − L ≤ −ɛ on V 0 .Pick a small ball B λ0 centered at λ 0 and a small disc ∆ w0 centered at w 0 such thatB 0 := B λ0 × ∆ w0 is relatively compact in V 0 . Then, according to Hartogs Lemma4.1.4, we have:(lim sup sup (L nj − L) ) ≤ sup(ϕ − L) ≤ −ɛ.j B 0 B 0This is impossible since, as we have seen before, we may find (λ 0 , r 0 e iθ 0( ) ∈ B 0 suchthat lim sup j Lnj (λ 0 , r 0 e iθ 0) − L(λ 0 ) ) = 0.⊓⊔62

Remark 4.2.8 Using standard techniques, one may deduce from the above Theoremthat the set of multipliers w for which the bifurcation current T bif is not a limit ofthe sequence d −n [Per n (w)] is contained in a polar subset of the complex plane.4.2.3 The case of neutral cycles in polynomial familiesWe return to the family ( P c,a)(c,a)∈C d−1 of degree d polynomials. Recall that P c,a isthe polynomial of degree d whose critical points are (0 = c 0 , c 1 , · · ·, c d−2 ) and suchthat P c,a (0) = a d (see subsection 2.1.2).We want to prove that, in this family, lim n d −n [Per n (w)] = T bif for |w| ≤ 1.Taking the results of the previous subsection into account (see Theorem 4.2.1), itremains to treat the case |w| = 1 and prove the following result due to Bassanelliand Berteloot (see [BB3]).Theorem 4.2.9 In the family of degree d polynomials lim n d −n [Per n (e iθ )] = T bif forany θ ∈ [0, 2π].We will follow a strategy similar to that used for proving Theorem 4.1.1. As weshall see, the proof would be rather simple if we would know that the bifurcationlocus is accumulated by hyperbolic parameters. This is however not the case whend ≥ 3 and is a source of technical difficulties (see the fourth step).Proof. We denote by λ the parameter in C d−1 (i.e. λ := (c, a)) and setL n (λ) := d −n ln |p n (λ, (e iθ ))|where the polynomials p n (λ, w) are those given by Theorem 2.2.1. We have to showthat the sequence (L n ) n converges to L in L 1 loc .We have already seen that(L n ) n is a uniformly locally bounded sequence of p.s.hfunctions on C d−1 . Since the family {P c,a } (c,a)∈C d−1 contains hyperbolic parameters,on which the L n (λ) = L + n (λ, eiθ ) it follows from Lemma 4.2.4 that the sequence(L n ) n does not converge to −∞. Thus, according to Theorem 4.1.3, we have toshow that L is the only limit value of the sequence (L n ) n for the L 1 loc convergence.Assume that (after taking a subsequence!) (L n ) n is converging in L 1 loc to ϕ. Toprove that the p.s.h functions ϕ and L coincide we shall check that they satisfy theassumptions of Lemma 4.1.5.First step: ϕ ≤ L.63

Since L + n (λ, eiθ ) converges to L in L 1 loc (see Lemma 4.2.4) and L n(λ) ≤ ̷L + n (λ, eiθ )we getϕ(λ 0 ) ≤ 1 ∫ϕ dm ≤ 1 ∫L dm|B ɛ | B ɛ|B ɛ | B ɛfor any small ball B ɛ centered at λ 0 . The desired inequality then follows by makingɛ → 0 since the function L is continuous (see Theorem 3.2.9).Second step: Supp dd c ϕ ⊂ Supp dd c L.Since there are no persistent neutral cycles in the family ( P c,a)(c,a)∈C d−1 , the hypersurfacesP er n (e iθ ) are contained in the bifurcation locus. This means that thefunctions L n are pluriharmonic on C d−1 \ Supp dd c L. The same is thus true for thelimit ϕ.Third step: for any λ 0 ∈ C d−1 there exists a complex line D through λ 0 suchthat ϕ = L on the unbounded component of D \ ( D ∩ Supp(dd c L) ) .By Theorem 2.3.5 we may pick a line D through λ 0 which meets infinity far fromthe cluster set of B i in P ∞ . This means that for any λ in the unbounded componentof D \ ( L ∩ Supp (dd c g) ) all critical point c i belongs to the super-attracting basinof ∞ and thus, λ is a hyperbolic parameter.This implies that L n (λ) = L + n (λ, e iθ ) and, by Lemma 4.2.4, ϕ(λ) = L(λ).Fourth step: ϕ = L on Supp dd c L.This is the most delicate part of the proof, it somehow proceeds by induction ond. To simplify the exposition, we will only treat the cases d = 2 and d = 3.When d = 2 the parameter space is C and the bifurcation locus is the boundarybM of the Mandelbrot set. The unbounded stable component ( M )c is hyperbolicand thus, as we saw in the last step, (ϕ − L) = 0 there. Since (ϕ − L) ≤ 0 and(ϕ − L) is u.s.c, this implies that ϕ = L on bM = Supp dd c LLet us stress that this ends the proof when d = 2 (the complex line D of the thirdstep is the parameter space itself in that case).We now assume that d = 3, the parameter space is then C 2 . Let us consider thesets U k of parameters which do admit an attracting k-cycle:U k := ⋃|w|

lim k d −k [Per k (0)] = T bif ), and the function (ϕ − L) is negative and upper semicontinuous,it suffices to prove that(ϕ − L) = 0 on all sets U k .Let us first treat the problem on a curve C := Per k (η) for |η| < 1 and show that(⋆) the sequence ϕ n | Cconverges uniformly to L| C on the stable components.We may assume that C is irreducible and desingularize it. This gives a onedimensionalholomorphic family (P π(u) ) u∈M . Keeping in mind that the elements ofthis family are degree 3 polynomials which do admit an attracting basin of periodk and using the fact that the connectedness locus in C 2 is compact (see Theorem2.3.5), one sees that the family (P π(u) ) u∈M enjoys the same properties than thequadratic polynomial family:1- the bifurcation locus is contained in the closure of hyperbolic parameters2- the set of non-hyperbolic parameters is compact in M.Exactly as for the quadratic polynomial family this implies that the sequence L nconverges in L 1 loc to L and the convergence is locally uniform on stable componentssince, as we already observed, the functions L n are pluriharmonic there.We now want to show that ϕ = L on any open subset U k . Again, as the stableparameters are dense and (ϕ − L) is u.s.c and negative, it suffices to show thatϕ = L on any stable component of U k . On such a component the functions L n arepluriharmonic and thus actually converge locally uniformly to ϕ. Then, (⋆) clearlyimplies that ϕ = L on Ω.⊓⊔4.3 Laminated structures in bifurcation loci4.3.1 Holomorphic motion of the Mandelbrot setWe work here in the moduli space Mod 2 of degree two rational maps which, as wesaw in section 2.1.3, can be identified to C 2 . Our aim is to show that the bifurcationlocus in the regionU 1 := {λ ∈ C 2 /f λ has an attracting fixed point}can be obtained by holomorphically moving the boundary bM of the Mandelbrotset . We remind that bM is the bifurcation locus of P er 1 (0) which is a complex linecontained in U 1 and can be identified to the family of quadratic polynomials.We will see simultaneoulsly that the bifurcation current is uniformly laminar in theregion U 1 . Let us first recall some basic facts about holomorphic motions.65

Definition 4.3.1 Let M be a complex manifold and E ⊂ Mbe any subset. A holomorphicmotion of E in M is a mapwhich satisfies the following properties:i) σ 0 = Id| Eσ : E × ∆ ∋ (z, u) ↦→ σ(z, u) =: σ u (z) ∈ Mii) E ∋ z ↦→ σ u (z) ∈ M is one-to-one for every u ∈ ∆iii) ∆ ∋ u ↦→ σ u (z) ∈ M is holomorphic for every z ∈ E.When the family of holomorphic discs in M enjoys good compactness properties,any holomorphic motion extends to the closure. In particular, when M is theRiemann sphere Ĉ, the Picard-Montel theorem combined with Hurwitz lemma easilyleads to the following famous extension statement. This result is usually calledλ-lemma since the ”time” parameter is denoted λ rather that t (see Lemm a 3.1.3).The main result of this subsection is the following. The prove we present here isdue to Bassanelli and Berteloot (see [BB2]).Theorem 4.3.2 Let Ω hyp be the union of all hyperbolic components of the Mandelbrotset M and ♥ the main cardioid. Let Bif 1 be the bifurcation locus in U 1 andT bif | U1 be the associated bifurcation current. Let µ 1 be the harmonic measure of M.There exists a continuous holomorphic motionsuch thatσ : (( Ω hyp \ ♥ ) ∪ bM ) × ∆ → U 1∫σ (bM × ∆) = Bif 1 and T bif | U1 =P er 1 (0)[σ(z, ∆)] µ 1 .In particular, Bif 1 is a lamination with µ 1 as transverse measure. Moreover, themap σ is holomorphic on ( Ω hyp \♥ ) ×∆ and preserves the curves P er n (w) for n ≥ 2and |w| ≤ 1.Proof.First step: Holomorphic motion of ( Ω hyp \ ♥ ) .The curve P er 1 (0) is actually the complex line λ 1 = 2. We will write (2, λ 2 ) =: zthe points of this line.Let us consider U n := {λ ∈ C 2 /f λ has an attracting cycle of period n} andΩ n := U n ∩ P er 1 (0). We recall that Ω hyp = ♥ ∪ ⋃ n≥2 Ω n.66

Let us also set U n,1 := U n ∩ U 1 . By the Fatou-Shishikura inequality, a quadraticrational map has at most two non-repelling cycles. Thus U n,1 ∩ U m,1 = ∅ whenn ≠ m and there exists a well defined holomorphic mapψ n : U n,1 → ∆ × ∆which associates to every λ ∈ U n,1 the pair (w n (λ), w 1 (λ)) where w n (λ) is the multiplierof the attracting n-cycle of λ and w 1 (λ) the multiplier of its attracting fixedpoint.The key point is the following transversality statement due to Douady and Hubbard(see also [BB1]):Lemma 4.3.3 the map ψ induces a biholomorphismψ n,j : U n,1,j → ∆ × ∆on each connected component U n,1,j of U n,1 .The connected components Ω n,j of Ω n coincides with U n,1,j ∩ P er 1 (0) and oneclearly obtains a holomorphic motion σ : ( Ω hyp \ ♥ ) × ∆ → U 1 by setting:σ(z, t) := (ψ n,j ) −1( w n (z), t )for any z ∈ Ω n,j .Second step: extension of σ to bM.The key point here is that σ(z, t) =: (α(z, t), β(z, t)) belongs to the complex lineP er 1 (t) which, according to Proposition 2.2.9, is given by the equation(t 2 + 1)λ 1 − tλ 2 − (t 3 + 1) = 0.Thus σ(z, t) is completely determined by β(z, t):α λ (t) = 11 + t 2 (tβλ (t) + t 3 + 2 ) , ∀t ∈ ∆. (4.3.1)We will now identify P er 1 (0) with the deleted Riemann sphere Ĉ \ {∞} andset β(∞, t) = ∞ for all t ∈ ∆. Then, the map β : ( {∞} ∪ ( Ω hyp \ ♥ )) × ∆ → Ĉis(clearly a holomorphic motion which, by Lemma 3.1.3, extends to the closure ofΩhyp \ ♥ ) . We thus obtain a continuous holomorphic motionβ : ( {∞} ∪ ( Ω hyp \ ♥ ) ∪ bM ) × ∆ → Ĉ.As, by construction, β(z, t) ≠ ∞ when z ≠ ∞, the identity 4.3.1 shows that σ(z, t) =(α(z, t), β(z, t)) extends to a continuous holomorphic motion of (( Ω hyp \ ♥ ) ∪ bM ) .67

Third step: laminarity properties.Let us show that T bif | U1 = ∫ [σ(z, ∆)] µ P er 1 (0) 1. According to the approximationformula given by Theorem 4.2.1 applied on P er 1 (0) we have:∑µ 1 = lim 2 −m δ σ(z,0) . (4.3.2)mz∈P er 1 (0)∩P er m(0)Let us set T := ∫ [σ(z, ∆)] µ P er 1 (0) 1. We have to check that T = T bif | U1 . Let φ bea (1, 1)-test form in U 1 . As the holomorphic motion σ is continuous, the functionz ↦→ 〈[σ(z, ∆)], φ〉 is continuous as well. Then, using 4.3.2 one gets∑〈T, φ〉 = lim 2 −m 〈[σ(z, ∆)], φ〉 = lim 2 −m 〈[P er m (0)], φ〉 (4.3.3)m mz∈P er n(0)∩P er m(0)where the last equality uses the fact that, according to Proposition 2.2.10, the curvesP er m (0) have no multiplicity in U 1 . Now the conclusion follows by using 4.3.3 andthe approximation formula of Theorem 4.2.1 in U 1 .By construction, the map σ is holomorphic on ( Ω hyp \ ♥ ) × ∆ and preserves thecurves P er n (w) for n ≥ 2 and |w| < 1. This extends to |w| = 1 by continuity.Using the continuity of σ, one easily sees that σ ( bM, ∆ ) is closed in U 1 andtherefore contains the support of ∫ P er 1 (0) [σ(z, ∆)] µ 1. By the above formula we thushaveBif1 = Supp ( T bif | U1)⊂ σ(bM, ∆).The opposite inclusion easily follows from the construction of σ:σ ( bM, ∆ ) is a limit of z m ∈ P er m (0) where m → +∞.COROLLARY: Closure of Bif in P2???.any point in⊓⊔Instead of using the basic λ-Lemma we could have use its far advanced generalizationdue to Slodkowski and get a motion on the full line P er 1 (0).Theorem 4.3.4 (Slodkowski λ-lemma) Let E ⊂ Ĉ be a subset of the Riemannsphere and σ : E × ∆ ∋ (z, t) ↦→ σ(z, t) ∈ Ĉ be a holomorphic motion. Then σextends to a holomorphic motion ˜σ of Ĉ. Moreover ˜σ is continuous on E × ∆ andz ↦→ ˜σ(z, t) is a K-quasi-conformal homeomorphism for K := 1+|t| . 1−|t|Our reference for quasi-conformal maps is the book [H] where one can also finda nice proof of Slodkowski theorem due to Chirka and Rosay.Using Slodkowski Theorem one may obtain further informations on the motiongiven by Theorem 4.3.2. We refer to our paper [BB2] for a proof.68

Theorem 4.3.5 Let σ : (( Ω hyp \ ♥ ) ∪ bM ) × ∆ → U 1 be the holomorphic motiongiven by Theorem 4.3.2. Then σ extends to a continuous holomorphic motion ˜σ :P er 1 (0) × ∆ −→ U 1 which is onto. All stable components in U 1 are of the form˜σ (ω × ∆) for some component ω in P er 1 (0). Moreover, the map λ ↦→ ˜σ(λ, t) is aquasi-conformal homeomorphism for each t and ˜σ is one-to-one on ( P er 1 (0) \ ♥ ) ×∆where ♥ is the main cardioid.Theorem 4.3.5 shows that non-hyperbolic components exist in U 1 if and onlyif such components exist within the quadratic polynomial family P er 1 (0). Let usunderline that, in relation with Fatou’s problem on the density of hyperbolic rationalmaps, it is conjectured that such components do not exist.4.3.2 Further laminarity statements for T bifThe following result is an analogue of Theorem 4.3.2 in the regionsU n = {λ ∈ Mod 2 /f λ has an attracting cycle of period n}.It shows, in particular, that the bifurcation current in Mod 2 is uniformly laminarin the regions U n . It has been established by Bassanelli and Berteloot in [BB2].Theorem 4.3.6 Let Bif n be the bifurcation locus in U n and T bif | Un be the associatedbifurcation current. Let Bifn c be the bifurcation locus in the central curve P er n (0)and µ c n be the associated bifurcation measure. Then, there exists a mapσ : Bifn c × ∆ −→ Bif n(λ, t) ↦−→ σ(λ, t)such that:1) σ (Bifn c × ∆) = Bif n2) σ is continuous, σ(λ, ·) is one-to-one and holomorphic for each λ ∈ Bifnc3) p n (σ(λ, t), t) = 0; ∀λ ∈ Bifn c, ∀t ∈ ∆4) the discs (σ(λ, ∆)) λ∈Bif cnare mutually disjoint.Moreover the bifurcation current in U n is given by∫T bif | Un = [σ (λ, ∆)] µ c nBif c nand, in particular, Bif n is a lamination with µ c nas transverse measure.69

The proof is similar to that of Theorem 4.3.6 but requires a special treatmentfor the extension problems since there is no λ-lemma available. The key is to usethe fact that, by construction, the starting motion σ : (∪ m≠n (U m ∩ U n )) × ∆ −→ U nsatisfies the following property:p n (σ(λ, t), t) = 0; ∀λ ∈ Bifn c , ∀t ∈ ∆.Such a motion is what we call a p n -guided holomorphic motion. Using Zalcmanrescaling lemma, one proves the following compactness property for guided holomorphicmotions. We stress that here, an holomorphic motion G is seen as familyof disjoints holomorphic discs σ and G t0 is the set of points σ(t 0 ).Theorem 4.3.7 Let p(λ, w) be a polynomial on C 2 × C such that the degree ofp(·, w) does not depend on w ∈ ∆. Let G be a p-guided holomorphic motion in C 2such that any component of the algebraic curve {p(·, t) = 0} contains at least threepoints of G t for every t ∈ ∆. Then, for any F ⊂ G such that F t0 is relatively compactin C 2 for some t 0 ∈ ∆, there exists a continuous p-guided holomorphic motion ̂F inC 2 such that F ⊂ ̂F and ̂F t0 = F t0 .The above Theorem plays the role of the λ-lemma in the proof of Theorem 4.3.6.We refer to the paper [BB2] for details.We will now end this section by quoting another laminarity result for T bif whichis due to Dujardin (see [Du])Theorem 4.3.8 Within the polynomial family of degree 3, the bifurcation currentis laminar outside the connectedness locus.We refer to section 2.3 for the discussion of the connectedness locus within polynomialfamilies.70

Chapter 5The bifurcation measure5.1 A Monge-Ampère mass related with strongbifurcations5.1.1 Basic propertiesSince the bifurcation current T bif of any holomorphic family (f λ ) λ∈Mof degree drational maps has a continuous potential L (see Definition 3.2.6 and Theorem 3.2.9),one may define the powers (T bif ) k := T bif ∧ T bif ∧ · · · ∧ T bif for any k ≤ m := dimM.We recall that for any closed positive current T , the product dd c L ∧ T is definedby dd c L ∧ T := dd c (LT ). In particular, (T bif ) m is a positive measure on M which isequal to the Monge-Ampère mass of the Lyapunov function L.Definition 5.1.1 Let (f λ ) λ∈Mbe a holomorphic family of degree d rational mapsparametrized by a complex manifold M of dimension m. The bifurcation measureµ bif of the family is the positive measure on M defined byµ bif = 1 m! (T bif) m = 1 m! (ddc L) mwhere T bif is the bifurcation current and L the Lyapunov function of the family.The following proposition is a direct consequence of the definition and the factthat µ bif has locally bounded potentials.Proposition 5.1.2 The support of µ bifµ bif does not charge pluripolar sets.is contained in the bifurcation locus andIt is actually possible to define the bifurcation measure in the moduli space Mod dof degree d rational maps and show that this measure has strictly positive and finitemass (see [BB1] Proposition 6.6). Although all the results we will present here aretrue in Mod d , we will restrict ourself to the technically simpler situation of holomorphicfamilies. The example we have in mind are the polynomial families and71

the moduli space Mod 2 which, in some sense, can be treated as a holomorphic family.In arbitrary holomorphic families, the measure µ bif can identically vanish. Moreover,it is usually quite involved to that µ bif > 0. Note however that this will followfrom standard arguments in polynomial families. The following simple observationalready shows that µ bif > 0 in M 2 (and more generally M d ), it has also its owninterest.Proposition 5.1.3 In any holomorphic family, all rigid Lattès examples belong tothe support of µ bif .Proof. The parameters corresponding to rigid Lattès examples are isolated. ByTheorem 1.3.5, these points correspond to strict minima of the Lyapunov functionL and therefore the Monge-Ampère measure (dd c L) m cannot vanish around them. ⊓⊔We end this subsection by showing that the activity currents have no selfintersection.This is a useful geometric information which, in particular, showsthat the activity of all critical points is a necessary condition for a parameter tobe in the support of the bifurcation masure. It was first proved by Dujardin-Favrein the context of polynomial families (see [DF] Proposition 6.9), we present here ageneral argument due to Gauthier ([G]).Theorem 5.1.4 Let (f λ ) λ∈Mbe any holomorphic family of degree d rational mapswith marked critical points. The activity currents T i satisfy T i ∧T i = 0. In particular,when m := dim M = 2d − 2 (or m = d − 1 for polynomial families) thenµ bif = T 1 ∧ T 2 ∧ · · · ∧ T mand Supp µ bifpoints.is contained in the intersection of the activity loci of the criticalThe proof is very close to that of a density statement which will be presentedin the next section, it combines the following potential-theoretic lemma with a dynamicalobservation.Lemma 5.1.5 Let u be a continuous p.s.h function on some open subset Ω in C 2 .Let Γ be the union of all analytic subsets of Ω on which u is harmonic. If the supportof dd c u is contained in Γ then dd c u ∧ dd c u vanishes on Ω.Proof. Let us set µ := dd c u ∧ dd c u. Let B r be an open ball of radius r whose closureis contained in Ω, we have to show that µ ( )B r2= 0.Denote by h the solution of the Dirichlet-Monge-Ampère problem with data uon bB r :72

h = u on the boundary of B rdd c h ∧ dd c h = 0 on B r (i.e. h is maximal on B r ).The function h is p.s.h and continuous on B r ( (see [BT]). As h is p.s.h maximaland coincides with the p.s.h function u on bB r , we have u ≤ h on B r . For any ɛ > 0we defineD ɛ := {λ ∈ B r2We will see that our assumption implies that/ 0 ≤ h(λ) − u(λ) ≤ ɛ}.Supp µ ∩ B r2 ⊂ D ɛ for all ɛ > 0. (5.1.1)Indeed, if γ is a complex curve in Ω on which u is harmonic then, the maximummodulus principle, applied to (h − u) on γ ∩ B r implies that h = u on γ.Then, as (h − u) is continuous on B r and Supp µ ⊂ Supp dd c u ⊂ Γ, we getSupp µ ∩ B r ⊂ {h = u}.Now, a result due to Briend-Duval (see[BD] or [DS] Théorème A.10.2) says thatµ (D ɛ ) ≤ Cɛ (5.1.2)where C only depends on u and B r . From 5.1.1 and 5.1.2 we deduce that µ ( )B r2= 0.⊓⊔We may now end the proof of Theorem 5.1.4.Proof. We only have to show that T i ∧ T i = 0, the remaining then follows from theidentity T bif = ∑ i T i (see Theorem 3.2.8).The statement is local and we may therefore assume that M = C k . Moreover, anelementary slicing argument allows to reduce the dimension to k = 2. We applythe above lemma with u = G λ (ĉ i (λ)) (see Lemma 3.2.7). We have to show thatthe support of dd c u = T i is accumulated by curves on which the critical point c i ispassive. These curves are of the form {fλ n(c i(λ) = c i (λ)} and their existence followsfrom Lemma 3.1.8.⊓⊔.An example, due to A.Douady, shows that the activity of all critical points isnot sufficent for a parameter to be in the support of the bifurcation measure. Wewill present this example in the next subsection (see 5.1.7).5.1.2 Some concrete familiesWe first discuss the case of the polynomial families introduced in subsection 2.1.2.We follow here the paper [DF] by Dujardin and Favre.73

Proposition( )5.1.6 The bifurcation measure µ bif of the degree d polynomial familyPc,a is a probability measure supported on the connectedness locus C. It(c,a)∈C d−1coincides with the pluricomplex equilibrium measure of C and its support is the Shilovboundary of C.Proof. Let us recall that the Green function of the polynomial P c,a is denoted g c,a .The connectedness locus C is a compact subset of C d−1 which coincides with theintersection ∩ 0≤i≤d−2 B i where B i is the set of parameters for which the orbit of thecritical point c i is bounded (see Theorem 2.3.5). As the support of the activity currentT i is contained in bB i we deduce that Supp µ bif ⊂ C from Theorem 5.1.4. All theremaining follows from the fact that µ bif = (dd c G) d−1 where G is the pluricomplexGreen function of C with pole at infinity:G := sup{u p.s.h / u − ln + max{|a|, |c k |} ≤ O(1), u ≤ 0 on C}an identity which we shall now prove.Let us first establish that µ bif = (dd c G) d−1 where G := max{g 0 , g 1 , · · ·, g d−2 }and g i := g c,a (c i ). We show by induction that T 0 ∧ T 1 ∧ · · · ∧ T l = (dd c G l ) l+1 for0 ≤ l ≤ d − 2 where G l := max{g 0 , g 1 , · · ·, g l }. This follows from the followingcomputation:T 0 ∧ T 1 ∧ · · · ∧ T l−1 ∧ T l = dd c ( g l (dd c G l−1 ) l) = dd c ( G l (dd c G l−1 ) l) =dd c ( G l−1 (dd c G l−1 ) l−1 ∧ dd c G l)= ddc ( G l (dd c M l−1 ) l−1 ∧ dd c G l)= (dd c G l ) l+1the second equality follows from g l = G L on the support of (dd c G l−1 ) l and thefourth from G l = G l−1 on the support of dd c G l , the last equality is obtained byrepeating the same arguments l − 1 times.It remains to show that G = G. The proof is standard and relies on the estimategiven by Proposition 2.3.7 and the fact that G is maximal outside C. We refer tothe paper [DF], Proposition 6.14 for more details.⊓⊔We will now present the example mentionned at the end of last subsection.Example 5.1.7 In the holomorphic family of degree 3 polynomials((1 + α1 )z + ( 1 2 + α 2)z 2 + z 3) α∈V 0where V 0 is a neighbourhood of the origin in C 2 the critical points are both active atthe origin (0, 0) but (0, 0) /∈ Supp µ bif .This family is a deformation of the polynomial P 0 := z + 1 2 z2 + z 3 . If V 0 issmall enough we have two marked critical points c 1 (α) and c 2 (α). The origin 0 isa parabolic fixed point for P 0 and we may assume that P α has two fixed points74

counted with multiplicity near 0 for all α ∈ V 0 . As P is real, its critical points arecomplex conjugate and both of them are attracted by the parabolic fixed point at0, moreover their orbits are not stationnary.We first show that (0, 0) /∈ Supp µ bif . When the fixed points of P α are distinct,we denote by m 1 (α) and m 2 (α) their multipliers. When this is the case, it turns outthat either |m 1 (α)| < 1 or |m 2 (α)| < 1 and thus one of the fixed points attracts acritical point. This can be seen by using the holomophic index fixed point formula.By Theorem 5.1.4, this implies that α /∈ Supp µ bif . We thus see that µ bif is supportedon the subvariety of parameters α for which the fixed point is double. ByProposition 5.1.2 this implies that µ bif vanishes near (0, 0).Let us now see that both critical points are active. We may assume that thefamily is parametrized by a disc D in C such that P α has two distinct fixed pointswhen α ≠ 0. Assume to the contrary that a critical point c(α) is passive. Then,after taking a subsequence, the sequence u n (α) := Pα n (c(α)) is uniformly convergingto u(α). Since the polynomial P 0 is real, its critical points are complex conjugateand must therefore both be attracted by the parabolic fixed point 0. Moreover, theirorbits are not stationnary. From this one easily deduce that the curve (α, u(α)) liesin the analytic set Z := {(α, z) ∈ D × P 1 / P α (z) = z}. This is impossible since, forsome α close to 0, the critical orbit should be attracted by a repelling fixed point.⊓⊔The situation in the moduli space Mod 2 is more complicated. We recall thatMod 2 can be identified to C 2 . Using the results which will be obtained in the lastsection of this chapter and the holomorphic motions constructed in section 4.3, it ispossible to show that the support of the bifurcation locus is not bounded.5.2 Density statementsWe show that the remarkable parameters introduced in subsection 3.1.3 accumulatethe support of the bifurcation measure.5.2.1 Misiurewicz parametersThe results given in this subsection are essentially due to Dujardin and Favre. Wepresent them in the setting of polynomial families and refer the reader to the originalpaper ([DF]) for a greater generality.Theorem 5.2.1 In the degree d polynomial family ( P c,a)(c,a)∈C d−1 let us define asequence of analytic sets by:W n0 ,···,n l:= ∩ l j=0 {P n jc,a(c j ) = P k(n j)c,a (c j )}75

where l ≤ d − 2 and k(n j ) < n j . Thenand W n0 ,···,n d−2lim nd−2 →∞ · · · lim n0 →∞is discrete.1d n d−2 +···+d n 0 [W n 0 ,···,n d−2] = µ bifProof. We treat the case d = 3 which is actually not very different from the generalcase.Let us first observe that W n0 ,n 1has codimension at least two and is containedin the connectedness locus which is compact (see Theorem 2.3.5). Thus W n0 ,n 1is adiscrete set.Applying a version of Theorem 4.1.1 suitably adapted to the family W n0yieldslimn 1 →∞ dn 1[W n0 ,n 1] = T 1 ∧ [W n0 ]where T 1 is the activity current of the critical point c 1 . By the same Theorem onehas lim n0 →∞ d n 0[W n0 ] = T 0 where T 0 is the activity current of c 0 and this, since T 1has continuous potentials, giveslimn 0 →∞ T 1 ∧ d n 0[W n0 ] = T 1 ∧ T 0 .The conclusion follows immediately since, according to Theorem 5.1.4, µ bif = T 0 ∧T 1 .⊓⊔An important consequence of the above result is that the support of the bifurcationmeasure is accumulated by Misiurewicz polynomials. An alternative proofof that fact will be given in the next subsection for arbitrary families. We refer to3.1.16 for a definition of Misiurewicz parameters.Corollary 5.2.2 In polynomial families, the support of the bifurcation measure iscontained in the closure of strongly Misiurewicz parameters: Supp µ bif ⊂ M iss.Proof. By the above Theoremlim · · · lim 1n d−2 →∞ n 0 →∞ d n d−2 + · · · + dn 0[∩ d−20 {P n jc,a (c j) = P n j−1c,a (c j )}] = µ bif . (5.2.1)Let us observe thatH j := {P n jc,a(c j ) = P n j−1c,a (c j )} = P reper nj + F ix jwhere, for parameters in P reper nj the critical point c j is strictly preperiodic to a(necessarily) repelling fixed point while c j is fixed for parameters in F ix j .Now Theorem 4.1.1 may ne rewritten as76

1limn j →∞ d [P reper n nj j] + α n jd [F ix j] = T n j j (5.2.2)but, as T j cannot charge the hypersurface F ix j , we must have αn jd n jlim nj →∞ 1d n j [P reper j ] = T j and 5.2.1 yieldslim · · · lim 1n d−2 →∞ n 0 →∞ d n d−2 + · · · + dn 0[∩ d−2j=0 P reper n j] = µ bif .→ 0. ThusThe conclusion follows immediately since ∩ d−2j=0 P reper n j⊂ M iss. ⊓⊔5.2.2 Shishikura or hyperbolic parametersWe aim here to show that the support of the bifurcation measure in Mod d is simultaneouslyaccumulated by Shishikura and hyperbolic parameters (see subsection3.1.3 for definitions):Supp µ bif⊂ Shi ∩ Hyp.It is worth emphasize that both statements will be deduced in the same wayfrom the following generalized version of Theorem 4.2.5.Theorem 5.2.3 Let µ bif be the bifurcation measure of a holomorphic family (f λ ) λ∈Mof rational maps. Let m denote the complex dimension of M. Let 0 < r ≤ 1. Thenthere exists increasing sequences of integers k 2 (n), ..., k m (n) such that:d −(n+k 2(n)+···+k m(n))∫m∧µ bif = lim [P ern m!(2π) mn (re iθ 1)] ∧ [P er kj (n)(re iθ j)] dθ 1 · · · dθ m .[0,2π] mj=2Moreover, we may assume that k j (n) ≠ k i (n) when i ≠ j.We will derive that result from Theorem 4.2.5 by simple calculus arguments withcurrents.Proof. For any fixed variety P er p (re iθp ), the set of θ ∈ [0, 2π] for which P er p (re iθp )shares a non trivial component with P er m (re iθ ) for some m ∈ N ∗ is at most countable.This follows from Fatou’s theorem on the finiteness of the set of non-repellingcycles. Thus, the wedge products [P er n1 (re iθ 1)] ∧ · · · ∧ [P er nm (re iθm )] make sensefor almost every (θ 1 , · · ·, θ m ) ∈ [0, 2π] m and the integrals ∫ [P er[0,2π] m n (re iθ 1)] ∧∧ mj=2 [P er k j (n)(re iθ j)] dθ 1 · · · dθ m are well defined.Next, we need the following formula which has been justified for q = 1 at theend of the proof of Theorem∫4.2.5. The proof is similar for q > 1 and we shall omitit. Recall that L r d−n 2πn (λ) := ln |p2π 0 n (λ, re iθ )| dθ.77

dd c L r n 1∧ · · · ∧ dd c L r n q= d−(n 1 +···+nq)(2π) q ∫[0,2π] q ∧ qk=1 [P er n k(re iθ k )]dθ1 · · · dθ q .To prove the convergence, we may replace M by C m since the problem is local.The conclusion is obtained by using Theorem 4.2.5, the above formula and the nextlemma inductively.Lemma 5.2.4 If S n → (dd c L) p for some sequence (S n ) n of closed, positive (p, p)-currents on M then dd c L r k(n) ∧ S n → (dd c L) p+1 for some increasing sequence ofintegers k(n).Let us briefly justify lemma 5.2.4. Let us denote by s n the trace measure of S n ,as M has been identified with C m this measure is given by s n := S n ∧ (dd c |z| 2 ) m−p .Since S n is positive, s n is positive as well. Let us consider the sequence (u k ) k definedby u k := L r k − L. We now that (u k) k converges pointwise to 0 (see remark 4.2.6) andis locally uniformly bounded (the function L is continuous). The positive currentS n may be considered as a (p, p) form whose coefficients are measures which aredominated by the trace measure s n . Thus, by the dominated convergence theorem,(L r k − L)S n = u k S n tends to 0 as k → ∞ and n is fixed. On the other hand, LS nconverges to LS because L is continuous. It follows that some subsequence L r k(n) S nconverges to LS.⊓⊔Corollary 5.2.5 In the moduli space Mod d the support of the bifurcation measureµ bif is contained in Shi ∩ Hyp.Proof. Use Remark 2.1.3 to work with families and then apply Theorem 5.2.3. For0 < r < 1 one gets Supp µ bif ⊂ Hyp and for r = 1 Supp µ bif ⊂ Shi ⊓⊔Let us end this subsection with a few remarks.We first notice that the above arguments yields a rather simple proof of the existenceof Shishikura maps, the original proof uses quasi-conformal surgery.Also, combining the above Corollary with Proposition 5.1.3 one sees that rigidLattès maps are accumulated by Shishikura’s maps or by hyperbolic maps. Thislast information apparently answers a question raised by Michel Herman.Theorem 5.2.3 remains true, with the same proof, if one replace the integrals by∫m∧[P er n (r 1 e iθ 1)] ∧ [P er kj (n)(r j e iθ j)] dθ 1 · · · dθ m[0,2π] mj=2where 0 < r j ≤ 1 for 1 ≤ j ≤ m. As a consequence, if α + ν = m and P α,ν is the setof parameters λ such that f λ has ν distinct attracting cycles and ν distinct neutralcycles Then Supp µ bif is contained in the closure of P α,ν .78

5.2.3 Shishikura parameters with chosen multipliersAs Theorem 4.2.9 shows, polynomial with a neutral cycle of a given multiplier aredense in the support of the bifurcation current. We believe that a similar propertyis still true for the bifurcation measure which means that Shishikura parameterswith arbitrarily fixed multipliers should be dense in the support µ bif . The followingresult goes in this direction.Theorem 5.2.6 Denote by p(f) (resp. s(f), c(f)) the number of distinct parabolic(resp. Siegel, Cremer) cycles of f ∈ Mod d . ThenSupp µ bif ⊂ {f ∈ Mod d / p(f) = p, s(f) = s and c(f) = c}for any triple of integers p, s and c such that p + s + c = 2d − 2.The proof is essentially based on Lemma 5.1.5 and Mañé-Sad-Sullivan theorem.More precisely we will use the followingLemma 5.2.7 Let E be a dense subset of [0, 2π]. Then for any holomorphic familyof degree d rational map (f λ ) Mthe setis dense in the bifurcation locus.∪ n ∪ θ∈E P er n (e iθ )Proof. Use Mañé-Sad-Sullivan theorem or Theorem 4.2.5 with r = 1.⊓⊔Let us now prove Theorem 5.2.6. We restrict ourself to Mod 2 . The general caserequires to use a slicing argument, we refer to [BB1] for details.Proof. Let E 1 and E 2 be two dense subsets of [0, 2π]. Let λ 0 be a point in thesupport of µ bif and U 0 be an arbitrarily small neighbourhood of λ 0 . By Lemma5.2.7, the support of the bifurcation current dd c L is accumulated by holomorphicdiscs contained in ∪ n ∪ θ∈E1 P er n (e iθ ). Among such discs, let us consider those whichgo through U 0 and pick one disc Γ 1 on which the Lyapunov function L is not harmonic.Such a disc exists since otherwise, according to lemma 5.1.5, the measureµ bif would vanish on U 0 . The bifurcation locus of (f λ ) Γ1is not empty and thus, toget a Shishikura parameter in U 0 with multipliers e iθ 1and e iθ 2where θ j ∈ E j , itsuffices to apply again Lemma 5.2.7 with the dense set E 2 to the family (f λ ) Γ1. ⊓⊔5.3 The support of the bifurcation measureIn this section we will show in which sense the support of the bifurcation measurein Mod d can be consider as a strong-bifurcation locus.79

5.3.1 A transversality resultTransversality statements play a very important role for understanding the structureof parameter spaces. We have alrady encounter such results like for instanceLemma 4.3.3 or the Fatou-Shishikura inequality. We refer to the fundamental workof Epstein [Eps2] for a general and synthetic treatement of transversality problemsin homorphic dynamics.All the results presented here are true in Rat d or in the moduli spaces Mod d . Forsimplicity we shall restrict ourself to the moduli space Mod 2 which will be treatedas a holomorphic family (f λ ) λ∈C 2 (see Theorem 2.2.8). We shall also assume, tosimplify the exposition, to have two marked critical points c j (λ) j = 1, 2.Assume that f 0 ∈ Mod 2 is strongly Misiurewicz. This means that there existstwo repelling cyclesand an integer k 0 ≥ 1 such thatfor j = 1, 2.C j (0) := {z j (0), · · ·, f n j−10 (z j (0))}f k 00 (c j (0)) = z j (0) but c j (0) /∈ C j (0)By the implicit function theorem, we may follow the cycles C j on a small ballB(0, r) centered at the origin. Writting C j (λ) := {z j (λ), · · ·, f n j−1λ(z j (λ))} the cyclescorresponding to the parameter λ ∈ B(0, r), we may define an important tool forstudying the parameter space near f 0 .Definition 5.3.1 The map χ : B(0, r) → C 2 defined byλ ↦→ ( f k 0λ (c j(λ)) − z j (λ) ) j=1,2is called activity map near the Misiurewicz parameter f 0 .As we shall see, thanks to the next result, the activity map will allow to transferinformations from the dynamical space of f 0 to the parameter space.Theorem 5.3.2 The activity map χ near a strongly Misiurewicz parameter f 0 islocally invertible.This Theorem was proved by Buff and Epstein (see [BE]) in the general settingof Rat d . In that case one has to assume that f 0 is not a flexible Lattès map (suchmaps do not exist in degree two). The proof of Buff and Epstein uses quadraticdifferentials thechniques. We shall prove here a weaker statement which is due toGauthier ([G]) and is sufficent for the applications we have in mind.80

Theorem 5.3.3 The activity map χ near a strongly Misiurewicz parameter f 0 islocally proper.The proof of that result is based on more classical arguments going back toSullivan (see also [vS] or [A]). The key point relies on the folowing Lemma.Lemma 5.3.4 All holomorphic curve contained in M iss consists of flexible Lattèsmaps.Proof. Assume to the contrary that (f λ ) λ∈Dis a holomorphic family parametrizedby a one-dimensional disc which is consisting of strongly Misiurewicz parametersand such that the f λ are not flexible Lattès maps.Then the Julia set of f λ coincides with P 1 for all λ ∈ D and, according to aTheorem of Mañé-Sad-Sullivan (see [MSS] Theorem B), there exists a quasiconformalholomorphic motion Φ : D × P 1 → P 1 which conjugates f λ to f 0 on P 1 . Let usdenote by µ λ the Beltrami form satisfying∂Φ λ∂¯z= µ λ ∂Φ λ. ∂zThere exists λ 1 ∈ D \{0} for which the support of µ λ 1has strictly positive Lebesguemeasure. Indeed, if this would not be the case, f λ would be holomorphically conjugatedto f 0 for all λ ∈ D. Then the Julia set of f λ1 carries an invariant line fieldand thus f λ1 is a flexible Lattès map (see [BM] Theorem VII. 22 and [Mc] corollary3.18). This is a contradiction. ⊓⊔We may now easily prove Theorem 5.3.3Proof. Let us first establish that at least one critical point must be activ. Assumeto the contrary that both critical points are passiv around f 0 . Then, according toLemma 3.1.17, f λ is strongly Misiurewicz for all λ ∈ B(0, r) after maybe reducing r.Cutting B(0, r) by a disc D passing through the origin we obtain, by lemma 5.3.4,a disc of flexible Lattès maps. Since this is impossible in Mod 2 (and by assumptionin other cases) we have reached a contradiction and proved that at least one criticalpoint, say c 1 , is activ at f 0 .The activity of c 1 means that χ −11 (0) has codimension one. The conclusion isobtained by repeating the argument on the hypersurface χ −11 (0). ⊓⊔5.3.2 The bifurcation measure and strong-bifurcation lociWe want to establish that the inclusion Supp µ bif ⊂ Shi∩M iss obtained in subsection5.2.2 is actually an equality. This is the reason for which we shall consider thesupport of the bifurcation measure as a strong-bifurcation locus. This is essentiallya consequence of the following result due to Buff and Epstein [BE].81

Theorem 5.3.5 In the moduli space Mod d the set of strongly Misiurewicz parametersis contained in the support of the bifurcation measure: M iss ⊂ Supp µ bif .Corollary 5.3.6 In the moduli space Mod d one has Supp µ bif= Shi = M iss.Proof. To simplify the presentation we will work in the degree 3 polynomial family(Pc,a)(c,a)∈C 2 . As usual we write λ the parameter (c, a) and c 1 (λ), c 2 (λ) the markedcritical points, the fact that in this setting c 1 = 0 does not play any role here.Assume that p 0 is a strongly Misiuerewicz polynomial. By definition, there existsan integer k 0 such that p k 00 (c j(0)) =: z j (0) is a repelling periodic point for j = 1, 2.To get lighter notations we whall assume that the z j (0) are fixed repelling points.We denote by z j (λ) the repellling fixed points which are obtained by holomorphicallymoving z j (0) on some neighbourhood of the origin and by w j (λ) the correspondingmultipliers. Observe that |w j (λ)| ≥ a > 1 on a sufficently small neighbourhoodof 0.The activity map χ (see definition 5.3.1) may be written: χ = (χ 1 , χ 2 ) whereχ j (λ) = p k 0λ(c j(λ)) − z j (λ).We will use here Theorem 5.3.2 and assume that χ is locally invertible at theorigin. It is possible adapt the proof for using the weaker transversality statementgiven by Theorem 5.3.3). For this one uses the fact that the sets obtained by rescalingthe ramifiction locus of χ are not charged by the measure µ bif and, thanks tosome Besicovitch covering argument, reduces the problem to some estimate similarto those which we will now perform in the invertible case. We refer to te papers[BE] and [G] for details.Let us denote by D 2 (0, ɛ) the bidisc centered at the origin and of multiradius(ɛ, ɛ) in C 2 . For ɛ small enough we may define a sequence of rescalingby setting δ n (x) := χ −1 (δ n : D 2 (0, ɛ) → Ω n). To prove the Theorem, it suffices to showx 1, x 2w 1 (0) w 2 (0)that µ bif (Ω n ) > 0 for all n. The crucial point of the proof is revealed by the followingcomputations.∫∫2µ bif (Ω n ) = Tbif ∫Ω 2 = δn ⋆ (T 1 + T 2 ) 2 ≥ δn ⋆ (T 1 ∧ T 2 ) =n D 2 (0,ɛ)D 2 (0,ɛ)∫= δn ⋆ [dd c g λ (c 1 (λ)) ∧ dd c g λ (c 2 (λ))]D 2 (0,ɛ)82

using the homogeneity property of the Green function(g λ (c j (λ)) = 3 −(k0+n) g λ pk 0 +nλ(c j (λ)) )one thus gets∫2 · 3 (k0+n) µ bif (Ω n ) ≥∫=D 2 (0,ɛ)δn⋆D 2 (0,ɛ)[dd c g λ ◦ p k 0+nλ(c 1 (λ)) ∧ dd c g λ ◦ p k 0+nλ(c 2 (λ)) ] =dd c g δn(x) ◦ p k 0+nδ n(x) (c 1(δ n (x))) ∧ dd c g δn(x) ◦ p k 0+nδ n(x) (c 2(δ n (x))).Let us express the quantities p k 0+nδ (c n(x) j(δ n (x)) by using the activity map χ. Bydefinition we have p k 0+nλ(c j (λ)) = p n λ (z j(λ) + χ j (λ)) and thus(p k 0+nδ (c n(x) j(δ n (x))) = p n δ n(x) z j (δ n (x)) +x )j.w j (0) nTo conclude, we momentarily admit the following()Claim: p n δ n(x)z j (δ n (x)) +x jw jis uniformly converging to some local biholomorphismψ j : C, 0 → P 1 , zj(0) n(0).As the Green function g λ (z) is continuous in (λ, z), the Claim implies that g δn(x)◦p k 0+nδ (c n(x) 2(δ n (x))) uniformly converges towards g 0 (ψ 0 (x j )) and our estimate yields∫lim infnµ bif (Ω n ) ≥D 2 (0,ɛ)dd c g 0 (ψ 1 (x 1 )) ∧ dd c g 0 (ψ 2 (x 2 )) =(∫) (∫)=dd c g 0 dd c g 0 > 0ψ 1 (D(0,ɛ))ψ 2 (D(0,ɛ))where the positivity of the last term follows from the fact that the repelling fixedpoints z j (0) = ψ j (0) belong to the Julia set of p 0 .It remains to justify the Claim. Let us write w j (λ) on the form w j (0) (1 + ɛ j (λ)).As ‖δ n (x)‖ ≤ C 1 one sees that (1 + ɛa nj (δ n (x))) n is uniformly converging to 1. Thenp n δ n(x)(z j (δ n (x)) +behaves like p n δ n(x)x )j= p nw j (0) n δ n(x)[z j (δ n (x)) +x jw j (δ n(x)) n ].[z j (δ n (x)) +]x jw j (δ n (x)) (1 + ɛ j(δ n n (x))) nNow let us linearize p λ near the repelling fixed points z j (λ). The linearizationholomorphically depends on the parameter λ and one gets local biholomorphismsψ j,λ such that ψ j,λ (0) = z j (λ) and83

p λ ◦ ψ j,λ (z) = ψ j,λ (w(λ)z) on B(0,ɛ). |w(λ)|Then the local biholomorphism ψ j of the Claim is simply ψ j,0 .⊓⊔Remark 5.3.7 By Theorem 5.1.4, (T 1 + T 2 ) 2 = 2T 1 ∧ T 2 and therefore the estimatein the proof of Theorem 5.3.5 becomes an equality:∫2 · 3 (k0+n) µ bif (Ω n ) =D 2 (0,ɛ)dd c g δn(x) ◦ p k 0+nδ n(x) (c 1(δ n (x))) ∧ dd c g δn(x) ◦ p k 0+nδ n(x) (c 2(δ n (x))).at (strongly) Mi-This allows to estimate the pointwise Hausdorff dimension of µ bifsiurewicz parameters.5.3.3 Hausdorff dimension estimatesHere, we will simply mention some further results obtained by T. Gauthier in histhesis (see [G]).Using the transversality map χ associated to Misiurewicz parameters (see Theorem5.3.3) it is possible to construct a ”transfer map” which copies some pieces ofthe dynamical plane into the parameter space. These tecniques have also been usedby Shishikura for proving that the boundary of the Mandelbrot set is of Hausdorffdimension 2 and by Tan Lei for proving that the bifurcation locus in any polynomialfamilies has also maximal Hausdorff dimension.This allows to relate the Hausdorff dimension of the strong -bifurcation locus withthe hyperbolic dimension of Misiurewicz parameters. Using Misiurewicz parameterswhose critical orbits are captured by hyperbolic sets with high Hausdorff dimensionthen gives the followingTheorem 5.3.8 The strong-bifurcation locus has full Hausdorff dimension.Combining this with Theorem 5.2.6 then yields theCorollary 5.3.9 Denote by p(f) (resp. s(f), c(f)) the number of distinct parabolic(resp. Siegel, Cremer) cycles of f ∈ Rat d . Let p, s and c be three integers such thatp + s + c = 2d − 2. Then the set{f ∈ Rat d / p(f) = p, s(f) = s and c(f) = c}is homogeneous and has maximal Hausdorff dimension 2(2d − 2).84

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