Parabolic implosion - from discontinuity to renormalization

Parabolic implosion
from discontinuity to renormalization
Mitsuhiro Shishikura
Congrès à la mémoire d'Adrien Douady
Institut Henri Poincaré, Paris, May 26-30, 2008
Discontinuity of Julia sets
“Periodicity’ in parameter space

Plan
Bifurcation of parabolic periodic points -- parabolic implosion
(saddle-node, intermittent chaos, periodic pts that bifurcate)
Local change of dynamics (“egg beater”) => Global effects
Main example: f 0 (z) =z + z 2 (∼ z 2 + 1 4
) and its perturbation
Discontinuity of Julia sets
Richness of bifurcated Julia sets
when perturbed to “wild direction’’
“Periodicity’’ in parameter space
Tools by Douady-Hubbard:
Fatou coordinates, Ecalle-Voronin cylinders, horn map, phase
Parabolic/Near-parabolic renormalization:
study of irrationally indifferent periodic points

Basic definitions
f
polynomial (or just holomorphic mapping for local problems)
fixed point
f(z 0 )=z 0 multiplier λ = f ′ (z 0 )
attracting
|λ| < 1
indifferent
|λ| =1
parabolic
α ∈ Q
λ = e 2πiα
irrationally indifferent
α ∈ R Q
repelling
|λ| > 1
filled-in Julia set
K f = {z ∈ C : {f n (z)} ∞ n=0
is bounded}
Julia set
(chaotic part)
J f = ∂K f = closure of {repelling periodic points}
We consider the bifurcation of a parabolic
fixed point and its global effect.

Tame and Wild directions
f ε (z) =z + z 2 + ... + ε
0 is parabolic for f 0
with λ 0 =1
Tame direction
safe side
Wild direction
rich side
narrow chance to be rich
Continuous change
when restricted to tame direction
for example, f(z) =e 2πiα z + ...
α ∈ R

Egg beater or Douady-Hubbard’s get-rich-quick scheme
f 0
perturbed into wild direction
Orbits from the left tend to
the fixed point
Imagine that a point in the box on the right
is: escaping point
(inverse image of) repelling periodic point
inverse image of critical point

Theorem (discontinity of Julia sets)
Let f ε (z) =z + z 2 + ε. Then
int K(f 0 )={z : f n → 0 uniformly in a neighborhood of z}
(parabolic basin).
Let {ε n } be a sequence such that ε n → 0 and { √ π
εn
} converges
modulo Z, i.e. for some integers k n ∈ Z,
( ) π
√ − k n = −β
εn
lim
n→∞
(for example, ε n =
Then
π2
(n−β)
). 2
J(f 0 ) J(< f 0 ,g β >) ⊂ lim inf
n→∞ J(f ε n
)
⊂ lim sup
n→∞
K(f εn ) ⊂ K(< f 0 ,g β >) K(f 0 )
where g β = lim n→∞ f k n
ε n
in int K(f 0 ), and
= “two generator dynamics” generated by f 0 and g β .

Pick a point w in the box on the right
which is escaping.
If a point z in the box on the left
arrives at this point, i.e. f k n
ε n
(z) =w,
then z ∈ int K(f 0 ) K(f εn ).
Hence int K(f 0 ) lim sup
n→∞
K(f εn ) ≠ ∅.
Pick a point w in the box on the right
which is an inverse image of a repelling
periodic point.
If a point z in the box on the left
arrives at this point, i.e. f k n
ε n
(z) =w,
then z ∈ J(f εn ) J(f 0 ).
Hence lim inf
n→∞ J(f ε n
) J(f 0 ) ≠ ∅.
The behavior of the critical point changes periodically. (“phase parameter”)
=⇒ the “periodicity” in the parameter space ( π √ ε
modulo Z matters).

Tools to analyze Egg beater dynamics
Difficulty: effects of perturbation on unboundedly high iterates of the map
Work in Fatou coordinates, which conjugate f to T : z ↦→ z +1
Take a croissant-shaped fundamental strip, which is bounded
by a curve l and its image f(l).
Glue l and f(l) by f to obtain a Riemann surface which is
isomorphic to C/Z.
Lift of the map to C/Z is a Fatou coordinate.
f
This part of dynamics is represented by
a horn map E f which is
– partially defined, non-linear
– continuous under perturbation.
Φ∞ attr
T Φ(w) rep
= w C/Z + 1 CΦ attr Φ rep C/Z C
This part of dynamics is represented by
an identification between attracting and
repelling Fatou coordinates which is
– an isomorphism
– sensitive wrt perturbation.

α z + . . .
e 2πiα z + λ . f(z) f. = 0 .
e e q z a 2 z + . . . a
Fatou coordinates ,
i ∈ N a
and Horn
i ≥ N
T (w) T (w) 2πi = qe = w 2πiα λ z = + 1O(z a
w + 1 1 2 ) 2 ≠ 0
α ∈ C {0} small
Φand Φ
| arg α|
map
a i ∈ N attr a attr Φ< i ≥ N rep π Φ rep C/Z C/Z mod CZ C E
= e 2πiα z + O(z 2 4
f0
α z + O(z 2 )
)
e 2πiα 0 (z) z + a 2 z 2 + . . .
F 0 (w) = w + 1 + o(1) w = − c f 0 (z) z + holomorphic f(z) O(z f ′ (0) 2 = ) = αe near 2πiα ∈e 2πiα R0 z , + Q
f (0) = 0 f ′ 0
α. (z) . . small holomorphic 0 (0) = near λ 0 f 0 (0) = 0 f 0 ′(0) = w
near 0 near mod Z E
λ = e 2πi ∞ p T (w) F 0 (w) w = 1w Φ1 + o(1) attr Φ w = rep C/Z − c λ
) iα = , fα 0
e(z) 2πiα small =
,
λz
α+ asmall
2 z 2 + . . . f 0 (z) = λz + a 2 z 2 + . . .
f0
w Cnear 0 near ∞ T
e f
q λ = mod 1 Za 2 ≠E0
0 R 0 f 0 R 0 f
2πiα , αf(z) αsmall
∈= a
C e {0} small and | arg α| < π i 2πiα ∈ Nz +
a i O(z
≥ N 2 )
f0
f f 0 R 0 f 0 R 0 f 0 (z) = z + O(z 2 0 (z) = e 2πi p q z + a 2 z 2 + . . . f
C {0} small and | arg α| < π 0 (z) = e 2πi p q z + a 2 z 2 + . . .
} small and | arg α| < π 4
)
{0} small f αf(z) ′ F 0
(0) ∈and R= e | 2πiα (w) = w + 1 2πiα arg4α| z + < f. 0
. π + o(1) w = − c near 0 near ∞ T (w) = w + 1 Φ attr
Q , α small . 4
z
f R 0 f 0 R 0 f 0 (z) = z + O(z 2 0 (z) = z + a 2 z 2 + mod . . . Z Ef 0 (z) = z + a 2 z 2 f0
4
E f + . . .
)
R Q
f Rf
fRf(z) Rf
= e 2πiβ z
Rf(z)
+ O(z 2 )
λ = e f(z) = e 2πiα z + O(z 2 )
Z
2πi p q
β = − 1 λ = 1 2 ≠ λ = e 2πi p q
Q αa ∈ i ∈C N f {0} a i ≥small N and λ | arg 1 α| a 2 < π 0 R 0 f 0 R 0 f 0 (z) = z + O(z ≠ 2 0)
α
mod Z f Rf Rf(z) 4= e 2πiβ z + O(z 2 ) β = − 1 ≥ N
α mod Z
N f(z) a i = ≥e 2πiα N
f ′ z + . . .
(0) e 2πiα f(z) = e 2πiα z + . . .
f Φ attr (z)
, α small
Φ attr (z)
attr (z)
+ 1 f(z) + o(1) = e
near 0 2πiα z w + = O(z− c Rf Rf(z) = e
a i ≥ N αF ∈ 0 (w) R = Qw + 1 + o(1)
2πiβ z + O(z
w = − c 2 ) β = −
near 0 1
near ∞ 2 )
) = w + 1 + o(1) w T (w) = − c α mod near Z ∞ T (w) = w + 1
near f(z) = 0 e
w 2πiα near z + O(z ∞
+ near 1 0Φ 2 ) T w
mod Z wE (w) = w + 1 Φ attr Φ rep C/Z C
Φ f0 attr (f(z)) attr near
α C {0} small and | arg α| < π Φ∞ rep Φ attr
T (f(z)) C/Z (w) = Cw w
+ attr
1(z) Φ+ attr 1
Φ rep rep (f(
= f 0 w
Z
f ′ (0) + 1
E= + e 2πiα o(1) , α small w = − c = Φ attr (z) + 1 Φ rep (f(z)) = Φ rep (z) + 1 Φ ... (f 0 (z)) = Φ ... (z) + 1
a i ∈ N a i ≥ Nf ′ (0) = near e
(z)) = Φ f0
rep (z) 2πiα , 0α small near ∞ T (w) = w + 1 Φ attr Φ rep C
Φnear rep ∞C/Z T (w) C = w + 1 Φ attr
w Φ rep attr (f(z)) C/Z C= Φ attr (z) 4 + Φ rep (f(z)) = Φ rep (z) + 1
E E f (z) = Φ attr attr Φ(f(z)) = Φ
R −1
f0
f 0 R
rep
att
α ∈ C {0} small and | argα α| ∈< C π
R α ∈ R Q 4
1 0 f 0 R 0 f 0 (z) = z + O(z E 2 {0} small and | arg
f (z) ) = Φ attr ◦ Φ −1
α| < π 0 f 0 (z) = z + E 0 f 0 R
O(z 2 0 f 0 (z) = z + O(z 2 )
)
F 0 (w) = w + 1 + o(1) w = − c f0 (z) = Φ attr ◦ Φ −1
rep χ f χ f (z) = z − 1 Rf = χ f ◦ E f
near40α
near ∞ T (w) = w + 1 Φ
w rep
0f 0 α ∈ R 0 mod
Q
α ∈ R Q
Rf(z) = e f P ◦ ϕ
E −1
1
f = P ◦ ϕ −1
f (z) = Φ attr ◦ Φ
Rf β = − 1 2πiβ f 0 (z) Z= z + O(z 2 )
z +
Rf f = EP O(z 2 )
Rf(z)
β =
=
−
e 12πiβ a
α
mod Z
i ∈ NRf(z) a i ≥ N= e 2πiβ z + O(z 2 ) β = − 1 α
mod Z
α mod z + Z O(z 2 ) β = − 1 f0 ◦ ϕ −1
α mod Z
a
α mod 1
=
f Z
i ∈ N a i ≥ N
f= ΦRf(z) f Q ◦ ϕ −1
attr Φ+ = 1e 2πiβ Φ rep
z + (f(z)) O(z 2 = ) where a i ∈ N
F f Q ◦ ϕ −1 0 (w) w + 1 + o(1) w = F− c Φ rep
β (z) = − 1 0 Rf + α1 mod Φ(f(z)) Z
attr (f(z)) 0 f= Q ◦ 0 Rϕ −1
= Φ 0 f 0 (z) = z + O(z
attr (z) + 1 Φ 2 )
rep 0 (w) = near w + 01 + near o(1) ∞w = T (w) − c = Φ(z) Φ rep + (z) 1 + 1 Φ ... (f 0 (z)) = Φ ... (
0 ∞ (−∞, = w near + 0 1Φnear attr ∞Φ rep T (w) C/Z = = Φ a 1 ±
0 ∞ f (−∞, 1 −1] P ◦ ϕ −1 w + C1 Φ attr
r(f(z)) = rep Φ(z) 1 Φ(f(z)) w −1]
Φ(z) 1
mod Z
attr (z) + 1 Φ rep (f(z)) = Φ rep (z) + w1 Φ(f(z)) = Φ(z) + 1
f0
mod Z E f0
rep(z) (z)) ttr ◦ + Φ= 1
−1
rep Φ attr
fE f0 (z) (z) Rf + = 1 ΦRf(z) attr Φ rep
◦(f(z)) Φ= −1 e 2πiβ rep = z Φ+ rep
O(z (z) 2 + ) 1 β = Φ
a ...
− 1 (f
2 ± α 0 (z)) mod = Z Φ ... (z) + 1
C R Z Q D H R Q C/Z C ∗ D ∗ C C D (−∞, −1]

}
f(z) = e z + O(z )
πi p small and
q z + a 2 z 2 | arg α| α< ∈ π C {0} small and | arg α| <
+
Perturbation
.
f(z) = e 2πiα . .
4
4
z + O(z 2 α ∈f R ′ (0) Q
) = e 2πiα , α small | arg α| < π 4
f ′ (0) = e 2πiα , α small
ff(z) = e 2πiα z + O(z 2 0
|
R
arg 0 f
α| 0 <
R π 0 f 0 (z))
= z + O(z 2 (wild direction)
)
α ∈ R Q
4
+ a 2 z 2 +
f 0 ′ . . .
(0) = e 2πiα a
, α small i ∈α N∈ Ca i ≥{0} Nf
small and | arg α| < π 4
α ∈ C {0} smallf and ′ (0) Rf | = arg e 2πiα α| Rf(z) , < α π small | arg α| < π
4
= e 2πiβ z + O(z 2 ) 4 β = − 1 ≥ N
a i ∈ N a i ≥ N
α ∈ C {0} small and | arg
α ∈
α|
R
<
π α
λ = 1 a 2 ≠ 0
F 0 (w) = w + Q1 + o(1) w = − c near 0 near ∞
α ∈ R Q α ∈ C
4
{0} small and w| arg α| < π 4
+ iα 1 + o(1) w = −F c 0 (w) near = mod w 0+ 1 Znear + Φo(1)
attr E∞ f0 (f(z)) w ET (w) f = −Φ = c attr w (z) + near 1+ 10 Φ attr
Φnear rep (f(z)) Φ∞ rep
= T C/Z (w) Φ rep = (
z + . . .
z
z
a i ∈ N a i ≥ N
f 0
Eα f
mod Z E
∈ R Q
f0 E
α f
a i ∈ N a i ≥ ∈ R Q
f N E iα z + O(z 2 f0 (z) = Φ attr ◦ Φ −1
0 R 0 f 0 R 0 f 0 (z) = z rep + O(z 2 )
)
R F 0 (w) = w + 1 + o(1) w = − c 0 f near 0 near
a 0 i
(z) ∈ N= z a+ i F
≥O(z f
N 0 2 ) R 0 f 0 R 0 f
a 0 (z) = z + O(z
i ∈ N a i ≥ N
0 (w) = w + 1 + w
mod
o(1)
Z
w
E
= − c 2 )
near 0 near ∞ T (w) = w
πiα f0
, α small | arg
F 0 (w) = w + 1 + o(1) w = −F c w
mod α| < Z π f Rf
f =
Rf(z)
P ◦ ϕ −1
= e 2πiβ z + O(z 2 ) β
E f0 E f
0 (w) near = w 0 1 + near o(1) ∞ w T = (w)
f 0 wR 0 f 0 R 0 f 0 (z) = z + O(z 2 − = c = − 1
4
α mod Z
Rf(z) = e 2πiβ z + O(z f 2 ) Rfβ = Rf(z) − 1 w + near 1 0Φ att n
) w
0} small modand Z | arg Ef f0 0
α| < R π α f mod = e 2πiβ Q Z z
◦ ϕ −1 + O(z 2 ) β = − 1 α mod Z
Φ
0 4f attr (f(z))
0 R 0 f 0 (z) mod= ZΦ z + attr
O(z E(z) 2 f0
+
) E1 f
Φ rep (f(z)) = Φ rep (z) + 1
= Φ attr (z) + 1 Φ rep
Φ attr (f(z)) (f(z)) = Φ= rep
Φ(z) attr 0
(z) + ∞ 1+ 1Φ (−∞, ... (f Φ rep 0 (z)) −1]
(f(z)) = Φ =
f f Rf Rf(z) = e 2πiβ ... (z) Φ rep + (z) 1 +
z + O(z 2 ) β = − 1 α mo
0 R 0 f 0 R 0 f 0 (z) = z + O(z f 2 0 )
R 0 f 0 R 0 f 0 (z) = z + O(z 2 )
f Rf Rf(z)
E e 2πiβ z + O(z 2 ) β = − 1 f0 (z) = Φ attr ◦ Φ −1
rep χ f χ f (z) =
C R Z Q
f Rf Rf(z) = e 2πiβ z
Φ
+ attr O(z f
(f(z))
2 ) Rf
=
β =
Φ
Rf(z) attr − 1 D H α R mod z − 1 1 Φ ... (f 0 (
Q Z Rf = χ
α
C/Z C ∗
attr ◦ Φ −1
rep χ f χ f
E(z) = z − 1 f0 (z) = Φ attr ◦Rf Φ −1
rep= χ f
χ f ◦ E f
χ f (z) ≃ = z − 1 ˜Rf = χ f ◦ E f
i ≥ N
α
(z) + 1 Φ rep (f(z)) = Φ rep (z) +
α mod = e 2πiβ Z α
z + O(z 2 ) β = − 1 Φ attr (f(z)) =
f =
Φ
P ◦ attr (z)
ϕ −1
first return map
+ 1 Φ rep (f(z)) = Φ rep (z) + 1 Φ ... (f 0 (z)) α
+ 1 + o(1) w = − c z ↦→ e 2πiz z ↦→ exp(2πiz)
+ o(1) (Im z → +∞) E f0 (z) = z + o(1) (Im z → +∞)
near 0E f0 near (z) = ∞Φ attr T ◦(w) Φ −1
rep = w + 1
Φ attr (f(z)) Φw
attr (z) + 1 Φ rep Φ attr (f(z)) = = Φ rep Φ attr (z) (z) + 1+ 1Φ ... Φ(f rep 0 (z)) (f(z)) = Φ = ... Φ(z) rep + (
depends Econtinuously f0 (z) = Φ attr ◦on
Φ −1
rep χ f χ f (z) = z − Φ attr Φ rep C/Z
f = Q ◦ ϕ −1
E f0 E f f Rf R 0 f R α f Rf Rf = χ f ◦ E f
e 2πiz Π : C/Z −→ ≃ Π C ∗ Π(z) = e
f 2πiz Π
= P ◦ ϕ −1 : C/Z −→ ≃ C ∗ αR 0 f 0 = Π ◦ E f0 ◦ Π −1
(after a suitable normalization)
−1
−1
−1

morphic near 0 f 0 (0) = 0 f ′ 0 (0) = λ
z + a 2 z 2 + . . .
πi p q z + a 2 z 2 + . . .
+ a 2 z 2 + . . .
λ = 1 a 2 ≠ 0
f 0
Lavaurs map
Lavaurs map
g β (z) =Φ −1
rep (Φ attr(z)+β)
describes how the orbits entering
attracting fundamental strip go out
from repelling fundamental strip.
πiα z + . . .
1near Φ attr ∞ T Φ(w) rep = C/Z w + 1 C Φ attr Φ rep C/Z C
πiα , α small
0} small and | arg α| < π 4z ↦→ z + β
or
1
α
mod Z
i ≥ N
∞+ 1 o(1)
rep(z) + T (w) 1+ = 1 w + = Φ(f(z)) 1− c Φ attr
w = ΦΦ(z) rep C/Z + 1 C
E f0
πiα z + O(z 2 )
g β = lim
n→∞ f k n
near 0 near ∞ T (w) = w + 1 Φ attr Φ rep C/Z C
n in int K(f 0)
if f n (z) =e 2πiα n
( z + ...,
1
)
− k n = −β,
lim
n→∞
α n
if f n (z) ( =z + z 2 + ε n ,
π
)
− k n = −β.
lim
n→∞
√
εn

Limit dynamics
f 0 ◦ g β = g β ◦ f 0 = g β+1
in K f0
〈f 0 ,g β 〉 = {f0 n ◦ gm β
: (m = 0 and n ≥ 0) or (m >0 and n ∈ Z)}
(when m>0, each element is defined in an open subset of C)
K(〈f 0 ,g β 〉)=C {z : ∃h = f n 0 ◦ gm β ∈〈f 0,g β 〉,h(z) ∈ C K f0 }
J(〈f 0 ,g β 〉) = closure of {repelling fixed points of h = f n 0 ◦ gm β ∈〈f 0,g β 〉}
J(f 0 ) J(〈f 0 ,g β 〉) ⊂ lim inf
n→∞ J(f n)
⊂ lim sup
n→∞
K(〈f 0 ,g β 〉)=J(〈f 0 ,g β 〉) =⇒
K(f n ) ⊂ K(〈f 0 ,g β 〉) K(f 0 )
lim K(f n) = lim J(f n)=J(〈f 0 ,g β 〉)
n→∞ n→∞
∃h = f n 0 ◦ g m β (∈ 〈f 0 ,g β 〉) has an attracting fixed point
=⇒ lim
n→∞ K(f n)=K(〈f 0 ,g β 〉) and
lim J(f n)=J(〈f 0 ,g β 〉)
n→∞

Further results
Discontinuity of straightening of polynomial-like mappings
(Douady-Hubbard, first published account on parabolic implosion)
Limit parameter space for z 2 1
+ c around c 0 =1/4 (wrt √ )
c−1/4
versus parameter space of 〈f 0 ,g β 〉 (Lavaurs)
Non-local connectivity of connectedness locus for real/complex cubic
polynomials (Lavaurs)
. . . . .
Parabolic/near-parabolic renormalization
Renormalization for germs holomorphic functions with parabolic or
near-parabolic fixed points (in wild direction)
S. with H. Inou: an invariant class of maps for the renormalization.
=> control on irrationally indifferent fixed points when
the continued fraction of the rotation number has large coefficients.
=> Buff-Cheritat’s result on quadratic Julia sets with positive area.

Return map and Renormalization
f
g
Rf
Renormalization
f Rf = first return map f (after rescaling)
If f is infinitely renormalizable, ...
f 0 = f f 1 = Rf 0 f 2 = Rf 1
h 0
g 0 g 1
h 1 h 2
˜g 0 ˜g 1
f Rf
as a “meta dynamical system”
on a space of dynamical systems
Often one expects a hyperbolic
f ↔ (α, f
dynamics 0 ) Rf(z) = e −2πi 1 α R α f 0 (z) R : (α,
f 0 α α ↦→ − 1 α mod Z R 0 R
R 0 contracting? R hyperbolic? (R α contrac
f = P ◦ ϕ −1
˜f 0 = ˜f ˜f1 = R ˜f 0
˜f2 = R ˜f 1
rigidity: weak conj. upgraded to nicer one
f = Q ◦ ϕ −1
0 ∞ (−∞, −1]

Near-parabolic Renormalization (cylinder renorm.)
(1) w = − c z
f
(z) = z + O(z 2 )
χ f
mod Z E f0 E f
near 0 near ∞ T (w) = w + 1 Φ attr Φ rep C/Z C
e 2πiβ z + O(z 2 ) β = − 1 Φα mod Z
attr (f(z)) = Φ attr (z) + 1 Φ rep (f(z)) = Φ rep (z) + 1 Φ ..
(z) + 1 Φ rep (f(z)) = Φ rep (z) + 1 Φ ... (f 0 (z)) = Φ ... (z) + 1
E f0 The (z) following = Φ attr ◦text Φ −1
rep is drawn χ f inχwhite.
f (z) = z − 1 α
1
ep χ f χ f (z) E f
= z − 1 Φ˜Rf attr χR f1 = ◦ E239
f
˜Rf Rf = χf ◦ E f ≃
αE f0 (z) = z + o(1) (Im z → +∞)
(Im z → +∞)
Π : C/Z
≃
F 0 (w) = w + 1 + o(1) w = − z
near 0 near ∞ T (
f 0 R 0 f 0 R 0 f 0 (z) = z + O(z 2 )
f Rf Rf(z) = e 2πiβ z + O(z 2 ) β = − 1 α mod Z
first return map
e −2π = 0.00186 . . .
h
−→ 1 =
C ∗ Exp ♯ ◦ E e 2πiα h ◦ ( Exp ♯) −1
Exp ♯ Exp ♯ (z) = e 2πiz Exp ♯ = R
: C/Z −→ ≃ α h
C ∗
Π Π(z) = e 2πiz Π : C/Z ≃
−→ C ∗ R 0 fR 0 = Π ◦ E f0 ◦ Π −1 R 0 f 0 Π ◦ E f0 ◦ −1 0 f 0 = Π ◦ E f0 ◦ Π −1 Rf = Π ◦ ˜Rf ◦ Π −1 = RΠ 0 f◦ 0 χ=
f ◦ E f
cylinders and Rf defined
R 0 f 0 = Exp ♯ ◦E f0 ◦ (Exp ♯ ) −1 R 0 f 0 = Exp ♯ ◦E f0 ◦ (Exp ♯ ) −1
where β = − 1 when 1 h ′′ (0) ≠ 0 and
(mod Z) or α = α sufficiently (m ∈small
N)
α m − β
f 0
R −→ f1
−1
R −→ f2
Exp ♯ : C/Z ≃ −→ C ∗ ,z↦→ e 2πiz
Rf is conjugate to the return map
on red croissant via repelling
Fatou coordinate and Exp ♯
R −→ f3
f = e 2πiα h, h = z + O(z 2 )
f ↔ (α,h)
R −→ . . .
˜Rf = χ f ◦ E
Rf = e −2πi 1 α h 1 , h 1 = z + O(z 2 )
Rf ↔ (− 1 α ,h 1)
R 0 f 0 = Π ◦ E f0 ◦ Π −

Theorem. (H. Inou and S.) There exists a class F 1 of maps
with a parabolic fixed point (non-degenerate) (iv) f ↦→and R 0 f is a“holomorphic.”
unique critical
point and a large number N such that the near-parabolic
renormalization R is defined on
univalent = holomorphic and injective
{e 2πiα h : α ∈ R, 0 < |α| ≤ 1 N ,h∈ F 1}
and hyperbolic. In fact, R can be considered as
R :(α,h) ↦→ (− 1 α mod Z, R αh),
(iii) If we write R 0 f = P ◦ ψ −1 , then ψ can be e
Theorem 2 The above statements hold for R α
exists an N such that the above holds for
α = 1
m + β
and α ↦→ − 1 α mod Z is expanding. The “fiber direction” F 1 is
in one to one correspondence with the Teichmüller space of
P (z) = z(1 + z) 2 and V , V ′
the punctured disk and the fiber map R α is holomorphic and
uniform contracting with respect to the Teichmüller distance.
with m ∈ N, β ∈ C
P (z) = z(1 + z) 2 P (0) = 0, P ′ (0) = 1
critical points: − 1 3
and −1; critical values: P (
Invariant class F 1 is characterized by (partial covering property.
η = 2
4
27 e2πη 4 27 e−2πη
U 4−
U 3+
η = 2 P slit
Dom(f)
cp
0
U 2
U 1
V slightly smaller domain than V ′
b
cv
f
0
c
C slit
f = P ◦ ϕ −1
−1

Applications (work in progress)
Theorem. Let f = e 2πiα h and ˆf = e 2πiα ĥ where h, ĥ ∈ F 1
(or z + z 2 ) and α is of high type (continued fraction coeffs ≥ N).
Then f and ˆf are quasiconformally conjugate on the closure of
the critical orbit. Moreover the conjugacy is C 1+γ -conformal on
the critical orbit with some γ > 0.
Compare with McMullen’s result on bounded type Siegel disks
Theorem. Let f = e 2πiα h where h ∈ F 1 (or z + z 2 ) and
α is of high type (continued fraction coefficientss ≥ N).
Then there exist open sets U n ∋ 0 and integers q n (n =0, 1,...)
such that f q n
is defined on U n and at most 3 to 1,
⋂ ∞
n=0 U n contains the critical orbit and consists of arcs (“hairs”)
which are disjoint from each other except at 0.
According to a discussion with Buff, Chéritat, Oversteegen,
quadratic Julia sets with non-Bruno, high-type rotation number
seem to be decomposable.

Merci!