Equilibrium measures for holomorphic endomorphisms of complex projective spaces
Volume 220 / 2013
Abstract
Let $f:\mathbb{P}\to\mathbb{P}$ be a holomorphic endomorphism of a complex projective space $\mathbb{P}^k$, $k\ge 1$, and let $J$ be the Julia set of $f$ (the topological support of the unique maximal entropy measure). Then there exists a positive number $\kappa_f>0$ such that if $\phi:J\to\mathbb{R}$ is a Hölder continuous function with $\sup(\phi)-\inf(\phi)<\kappa_f$, then $\phi$ admits a unique equilibrium state $\mu_\phi$ on $J$. This equilibrium state is equivalent to a fixed point of the normalized dual Perron–Frobenius operator. In addition, the dynamical system $(f,\mu_\phi)$ is K-mixing, whence ergodic. Proving almost periodicity of the corresponding Perron–Frobenius operator is the main technical task of the paper. It requires producing sufficiently many “good” inverse branches and controling the distortion of the Birkhoff sums of the potential $\phi$. In the case when the Julia set $J$ does not intersect any periodic irreducible algebraic variety contained in the critical set of $f$, we have $\kappa_f=\log d$, where $d$ is the algebraic degree of $f$.
