My title - Departamento de Matemática da Universidade do Minho

11
RECORRÊNCIAS 76
Poincaré recurrence theorem (topologic version). Let f : X → X be a continuous
transformation of a separable metrizable topological space X. The support of any invariant Borel
probability measure µ is contained in the closure of the set of recurrent points, namely
supp (µ) ⊂ Rec f .
If, in particular, f admits an invariant measure µ which is diffuse (i.e. gives positive measure to
any nonempty open set) then the set of recurrent points is dense in X, namely
Rec f = X.
Observe that if f is a homeomorphism, then the same applies to Rec f −1, and the support of
any invariant Borel probability measure is contained in the closure of Rec f ∩Rec f −1.
If you don’t like the above proof, here is another, perhaps more elemenary, of the last statement.
Alternative proof. Assume that the continuous map f : X → X preserves a diffuse Borel
probability measure µ. For each n ≥ 1, let
R n = { x ∈ X s.t. ∃ k ≥ 1 s.t. d(f k (x), x) < 1/n }
be the set of ”1/n-recurrent” points. Of course, Rec f = ∩ ∞ n=1R n . The sets R n are clearly open.
To show that Rec f is dense we must show that each R n is, since then the Baire theorem implies
that also their countable intersection is dense. So, take any nonempty ball B with diameter
< 1/n. Its inverse images f −1 (B), f −2 (B), f −3 (B),... have all the same measure µ(B) > 0. Since
µ(X) = 1, they cannot be disjoint. There follows that there exist k > 0 and n ≥ 0 such that
f −(n+k) (B) ∩ f −n (B) ≠ 0, and this implies that B contains a 1/n-recurrent point (for a point x in
the intersection has both images f n (x) and f n+k (x) = f k (f n (x)) in B, hence at distance < 1/n).
Since B was arbitrary, this proves that each R n is dense, and Baire theorem implies that Rec f is
dense too.
Example. Rotations of the circle preserve the Lebesgue probability measure. Hence, almost any
point x + Z is recurrent for the rotation R α : x + Z ↦→ x + α + Z. If α is rational this is trivial, since
all points are periodic. When α is irrational, this says that for any ε > 0 there exist an infinity of
times q ∈ N that
d(x + Z, x + qα + Z) < ε
hence for any ε > 0, there exist an infinity of rationals p/q such that
|qα − p| < ε i.e.
∣ α − p q ∣ < ε q
We know from continued factions that the estimate is indeed better, since ε/q may be substituited
by 1/q 2 .
11.4 Conjunto não-errante
Por volta dos anos trinta, George D. Birkhoff teve a ideia de dividir o espaço dos estados de um
sistema dinâmico em duas classes de pontos com dinâmicas qualitativamente distintas.
O ponto x é errante 26 se admite uma vizinhança disjunta de todas as suas iteradas, i.e. se
existe um aberto U que contém x tal que U ∩ f n (U) = ∅ para todo tempo n ≥ 1. O ponto x não
é errante se para toda vizinhança U de x existe um tempo n ≥ 1 tal que f n (U) ∩ U ≠ ∅.
O conjunto não-errante NW f (do inglês “non-wandering set”) é o conjunto dos pontos x que não
são errantes. A ideia informal é que conjunto não-errante é onde acontece a dinâmica interessante,
enquanto o conjunto errante é o conjunto dos pontos que a dinâmica esquece.
26 A palavra grega por “errante”, ou seja, “vagabundo”, “que vagueia ao acaso”, era πλανητης, ou seja, planeta.