Discrete Holomorphic Local Dynamical Systems

222 Tien-Cuong Dinh and Nessim Sibony
Exercise 1.104. Show that if a Borel set A satisfies μ(A) > 0, then μ( f n (A))
converges to 1.
Exercise 1.105. Show that sup ψ In(ϕ,ψ) with ψ smooth �ψ�∞ ≤ 1 is equal to
�ϕ� L 1 (μ) . Deduce that there is no decay of correlations which is uniform on �ψ�∞.
Exercise 1.106. Let V1 := {ψ ∈ L 2 0 (μ), Λ(ψ) =0}. Show that V1 is infinite
dimensional and that bounded functions in V1 are dense in V1 with respect to
the L 2 (μ)-topology. Using Theorem 1.82, show that the only eigenvalues of Λ are
0 and 1.
Exercise 1.107. Let ϕ be a d.s.h. function as in Corollary 1.88. Show that
�ϕ + ···+ ϕ ◦ f n−1 � 2
L 2 (μ) − nσ 2 + γ = O(d −n ),
where γ := 2∑n≥1 n〈μ,ϕ(ϕ ◦ f n )〉 is a finite constant. Prove an analogous property
for ϕ Hölder continuous.
1.7 Entropy, Hyperbolicity and Dimension
There are various ways to describe the complexity of a dynamical system. A basic
measurement is the entropy which is closely related to the volume growth of the
images of subvarieties. We will compute the topological entropy and the metric entropy
of holomorphic endomorphisms of P k . We will also estimate the Lyapounov
exponents with respect to the measure of maximal entropy and the Hausdorff
dimension of this measure.
We recall few notions. Let (X,dist) be a compact metric space where dist is a
distance on X. Letg : X → X be a continuous map. We introduce the Bowen metric
associated to g. For a positive integer n, define the distance distn on X by
distn(x,y) := sup
0≤i≤n−1
dist(g i (x),g i (y)).
We have distn(x,y) > ε if the orbits x,g(x),g 2 (x),... of x and y,g(y),g 2 (y),... of y
are distant by more than ε at a time i less than n. In which case, we say that x,y are
(n,ε)-separated.
The topological entropy measures the rate of growth in function of time n, of
the number of orbits that can be distinguished at ε-resolution. In other words, it
measures the divergence of the orbits. More precisely, for K ⊂ X, not necessarily invariant,
let N(K,n,ε) denote the maximal number of points in K which are pairwise
(n,ε)-separated. This number increases as ε decreases. The topological entropy of
g on K is
1
ht(g,K) := suplimsup
ε>0 n→∞ n logN(K,n,ε).