Equilibrium measures for holomorphic endomorphisms of complex projective spaces

Mariusz Urbański ¹   ; Anna Zdunik ²

¹ Department of Mathematics University of North Texas Denton, TX 76203-1430, U.S.A.
² Institute of Mathematics University of Warsaw 02-097 Warszawa, Poland

Fundamenta Mathematicae, Tome 220 (2013) no. 1, pp. 23-69

Résumé

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 $f$.

Détail

DOI : 10.4064/fm220-1-3

Keywords: mathbb mathbb holomorphic endomorphism complex projective space mathbb julia set topological support unique maximal entropy measure there exists positive number kappa phi mathbb lder continuous function sup phi inf phi kappa phi admits unique equilibrium state phi equilibrium state equivalent fixed point normalized dual perron frobenius operator addition dynamical system phi k mixing whence ergodic proving almost periodicity corresponding perron frobenius operator main technical task paper requires producing sufficiently many inverse branches controling distortion birkhoff sums potential phi julia set does intersect periodic irreducible algebraic variety contained critical set have kappa log where algebraic degree nbsp