Central Limit Theorem and regularity of the maximal entropy measure

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 $\mathbb{C}\mathbb{P}(1)$.