Entropy of meromorphic maps acting on analytic sets

Abstract

Let $f:X\to X$ be a dominating meromorphic map on a compact Kähler manifold $X$ of dimension $k$. We extend the notion of topological entropy $h_{\mathrm{top}}^l(f)$ for the action of $f$ on (local) analytic sets of dimension $0\le l\le k$. For an ergodic probability measure $\nu$, we extend similarly the notion of measure-theoretic entropy $h_{\nu }^l(f)$. Under mild hypothesis, we compute $h_{\mathrm{top}}^l(f)$ in term of the dynamical degrees of $f$. In the particular case of endomorphisms of ${\mathbb{P}}^2$ of degree $d$, we show that $h_{\mathrm{top}}^1(f)=\log d$ for a large class of maps but we give examples where $h_{\mathrm{top}}^1(f)\ne \log d$.