Central Limit Theorem and regularity of the maximal entropy measure
- Speaker(s)
- Christophe Dupont
- Affiliation
- Université Paris-Sud 11
- Date
- Feb. 27, 2009, 10:15 a.m.
- Room
- room 5840
- Seminar
- Seminar of Dynamical Systems Group
Let $f$ be an holomorphic endomorphism of $\mathbb{C}\mathbb{P}(2)$ of
degree $d \geq 2$. It is known that the measure of maximal entropy $\mu$
is absolutely continuous with respect to the Lebesgue measure when its
Lyapunov exponents are minimal equal to $\log \sqrt d$. That can be
obtained by classical arguments, by disintegrating $\mu$ along suitable
dilated partitions. We propose here a new proof based on the Central
Limit Theorem for the observable $\log Jac f$. We prove the CLT for that
unbounded function by constructing a weak Bernoulli coding map, we
extend for that purpose a technique introduced by Przytycki, Urba\'nski
and Zdunik for rational fractions on $\mathbb{C}\mathbb{P}(1)$.