Discrete Holomorphic Local Dynamical Systems

230 Tien-Cuong Dinh and Nessim Sibony
Consider the second case with multi-indices i ∈ I. Letqdenote the integer part
of λ n and Wα the projection of Γn ∩ Uα on Pk by Π0. Observe that the choice of the
open sets Ui implies that f q is injective on Wα. Therefore,
∑ α
�
Γn∩Uα
≤ ∑
α Wα
�
= ∑
α
�
≤ ∑ α
Π ∗
i1 (ωFS) ∧ ...∧ Π ∗
i (ωFS)
k
�
( f i1 ∗
) (ωFS) ∧ ...∧ ( f ik ∗
) (ωFS)
Wα
( f q ) ∗� ( f i1−q ) ∗ (ωFS) ∧ ...∧ ( f i k−q ) ∗ (ωFS) �
P k( f i1−q ) ∗ (ωFS) ∧ ...∧ ( f i k−q ) ∗ (ωFS).
Recall that the number of open sets Uα is bounded by d kρn . So, the last sum is
bounded by
d kρn d (i1−q)+···+(i k−q) ≤ d kρn d k(n−q) � d k(1+ρ−λ )n .
Finally, since the number of multi-indices i is less than n k , we deduce from the
above estimates that
k!volume(Γn ∩ Vn) � n k d (k−1+λ )n + n k d k(1+ρ−λ )n .
This contradicts the above bound from below of volume(Γn ∩ Vn). ⊓⊔
The remaining part of this paragraph deals with Lyapounov exponents associated
to the measure μ and their relations with the Hausdorff dimension of μ. Results
in this direction give some information about the rough geometrical behaviour of
the dynamical system on the most chaotic locus. An abstract theory was developed
by Oseledec and Pesin, see e.g. [KH]. However, it is often difficult to show that
a given dynamical system has non-vanishing Lyapounov exponents. In complex
dynamics as we will see, the use of holomorphicity makes the goal reachable. We
first introduce few notions.
Let A be a linear endomorphism of R k . We can write R k as the direct sum ⊕Ei
of invariant subspaces on which all the complex eigenvalues of A have the same
modulus. This decomposition of R k describes clearly the geometrical behaviour of
the dynamical system associated to A. An important part in the dynamical study
with respect to an invariant measure is to describe geometrical aspects following
the directional dilation or contraction indicators.
Consider a smooth dynamical system g : X → X and an invariant ergodic probability
measure ν. Themapg induces a linear map from the tangent space at x to
the tangent space at g(x). This linear map is given by a square matrix when we fix
local coordinates near x and g(x). Then, we obtain a function on X with values in
GL(R,k) where k denotes the real dimension of X. We will study the sequence of
such functions associated to the sequence of iterates (g n ) of g.