Discrete Holomorphic Local Dynamical Systems

228 Tien-Cuong Dinh and Nessim Sibony
and
h − ν (g,x) := sup
δ>0
1
liminf−
n→∞ n logν(Bg n(x,δ)).
Theorem 1.114 (Brin-Katok). Let g : X → X be a continuous map on a compact
metric space. Let ν be an invariant probability measure of finite entropy. Then,
hν(g,x) =h − ν (g,x) and hν(g,g(x)) = hν(g,x) for ν-almost every x. Moreover,
〈ν,hν(g,·)〉 is equal to the entropy hν(g) of ν. In particular, if ν is ergodic, we have
hν(g,x)=hν(g) ν-almost everywhere.
One can roughly say that ν(B g n(x,δ)) goes to zero at the exponential rate e −hν(g)
for δ small. We can deduce from the above theorem that if Y ⊂ X is a Borel set with
ν(Y ) > 0, then ht(g,Y) ≥ hν(g). The comparison with the topological entropy is
given by the variational principle [KH, W].
Theorem 1.115 (variational principle). Let g : X → X be a continuous map on a
compact metric space. Then
suphν(g)=ht(g),
where the supremum is taken over the invariant probability measures ν.
Newhouse proved in [NE] thatifg is a smooth map on a smooth compact manifold,
there is always a measure ν of maximal entropy, i.e. hν(g) =ht(g). One of
the natural question in dynamics is to find the measures which maximize entropy.
Their supports are in some sense the most chaotic parts of the system. The notion
of Jacobian of a measure is useful in order to estimate the metric entropy.
Let g : X → X be a measurable map as above which preserves a probability
measure ν. Assume there is a countable partition (ξi) of X, such that the map g
is injective on each ξi. The Jacobian Jν(g) of g with respect to ν is defined as
the Radon-Nikodym derivative of g ∗ (ν) with respect to ν on each ξi. Observe that
g ∗ (ν) is well-defined on ξi since g restricted to ξi is injective. We have the following
theorem due to Parry [P].
Theorem 1.116 (Parry). Let g, ν be as above and Jν(g) the Jacobian of g with
respect to ν.Then
�
hν(g) ≥ logJν(g)dν.
We now discuss the metric entropy of holomorphic maps on P k . The following
result is a consequence of the variational principle and Theorems 1.108 and 1.112.
Corollary 1.117. Let f be an endomorphism of algebraic degree d ≥ 2 of P k .Letν
be an invariant probability measure. Then hν( f ) ≤ k logd. If the support of ν does
not intersect the Julia set Jp of order p, then hν( f ) ≤ (p − 1)logd. In particular,
if ν is ergodic and hν( f ) > (p − 1)logd, then ν is supported on Jp.
In the following result, the value of the metric entropy was obtained in [BD2,S3]
and the uniqueness was obtained by Briend-Duval in [BD2]. The case of dimension
1 is due to Freire-Lopès-Mañé[FL] and Lyubich [LY].