Stochastics and thermodynamics for equilibrium measures of holomorphic endomorphisms on complex projective spaces

It was proved by Urbański and Zdunik (Fund Math 220:23–69, 2013) that for every holomorphic endomorphism $f:{{\mathbb { P}}}^k\rightarrow {{\mathbb { P}}}^k$ of a complex projective space ${{\mathbb { P}}}^k,k\ge 1$ , there exists a positive number $\kappa _f>0$ such that if $J$ is the Julia set of $f$ (i.e. the support of the maximal entropy measure) and $\phi :J\rightarrow {\mathbb R}$ is a Hölder continuous function with $\sup (\phi )-\inf (\phi )<\kappa _f$ (pressure gap), then $\phi $ admits a unique equilibrium state $\mu _\phi $ on $J$ . In this paper we prove that the dynamical system ( $f,\mu _\phi $ ) enjoys exponential decay of correlations of Hölder continuous observables as well as the Central Limit Theorem and the Law of Iterated Logarithm for the class of these variables that, in addition, satisfy a natural co-boundary condition. We also show that the topological pressure function $t\mapsto P(t\phi )$ is real-analytic throughout the open set of all parameters $t$ for which the potentials $t\phi $ have pressure gaps.