Discrete Holomorphic Local Dynamical Systems

Dynamics in Several Complex variables 227
The entropy of ξ is the average of I ξ :
�
H(ξ ) :=
I ξ (x)dν(x)=−∑ν(ξi)logν(ξi).
It is useful to observe that the function t ↦→−tlogt is concave on ]0,1] and has the
maximal value e−1 at e−1 .
Consider now the information obtained if we measure the position of the orbit
x,g(x),...,gn−1 (x) relatively to ξ . By definition, this is the measure of the entropy
of the partition generated by ξ ,g−1 (ξ ),...,g−n+1 (ξ ), which we denote by
�n−1 i=0 g−i (ξ ). The elements of this partition are ξi1 ∩ g−1 (ξi2 ) ∩ ...∩ g−n+1 (ξin−1 ).
It can be shown [W]that
1
hν(g,ξ ) := lim
n→∞ n H
� n−1
�
exists. The entropy of the measure ν is defined as
i=0
hν(g) := suphν(g,ξ
).
ξ
g −i (ξ )
Two measurable dynamical systems g on (X,F ,ν) and g ′ on (X ′ ,F ′ ,ν ′ ) are
said to be measurably conjugate if there is a measurable invertible map π : X → X ′
such that π ◦ g = g ′ ◦ π and π∗(ν) =ν ′ . In that case, we have hν(g) =h ν ′(g ′ ).
So, entropy is a conjugacy invariant. Note also that hν(g n )=nhν(g) and if g is
invertible, hν(g n )=|n|hν(g) for n ∈ Z. Moreover, if g is a continuous map of a
compact metric space, then ν ↦→ hν(g) is an affine function on the convex set of
g-invariant probability measures [KH, p.164].
We say that a measurable partition ξ is a generator if up to sets of measure
zero, F is the smallest σ-algebra containing ξ which is invariant under g n ,n ∈ Z.
A finite partition ξ is called a strong generator for a measure preserving dynamical
system (X,F ,ν,g) as above, if up to sets of zero ν-measure, F generated
by � ∞ n=0 g −n (ξ ). The following result of Kolmogorov-Sinai is useful in order to
compute the entropy [W].
Theorem 1.113 (Kolmogorov-Sinai). Let ξ be a strong generator for the
dynamical system (X,F ,ν,g) as above. Then
hν(g)=hν(g,ξ ).
We recall another useful theorem due to Brin-Katok [BK] which is valid for
continuous maps g : X → X on a compact metric space. Let B g n(x,δ) denote the
ball of center x and of radius δ with respect to the Bowen distance distn. We call
B g n(x,δ) the Bowen (n,δ)-ball. Define local entropies of an invariant probability
measure ν by
hν(g,x) := sup
δ>0
�
limsup−
n→∞
1
n logν(Bg n(x,δ))