On the support of measures of large entropy for polynomial-like maps

Abstract

Let $f$ be a polynomial-like map with dominant topological degree $d_t\ge 2$ and let $d_{k-1}<d_t$ be its dynamical degree of order $k-1$. We show that the support of every ergodic measure whose measure-theoretic entropy is strictly larger than $\log \sqrt{d_{k-1}{d}_t}$ is supported on the Julia set, i.e., the support of the unique measure of maximal entropy $\mu$. The proof is based on the exponential speed of convergence of the measures $d_t^{-n}(f^n{)}^{\ast }{\delta }_a$ towards $\mu$, which is valid for a generic point $a$ and with a controlled error bound depending on $a$. Our proof also gives a new proof of the same statement in the setting of endomorphisms of ${\mathbb{P}}^k(\mathbb{C})$ - a result due to de Thélin and Dinh - which does not rely on the existence of a Green current.