Entropy of meromorphic maps acting on analytic sets

TL;DR

This paper extends the concept of topological and measure-theoretic entropy to the action of meromorphic maps on analytic sets within compact Kähler manifolds, linking entropy to dynamical degrees and providing specific results for endomorphisms of the projective plane.

Contribution

It introduces a generalized notion of entropy for meromorphic maps acting on analytic sets and relates it to dynamical degrees, with explicit computations for certain endomorphisms.

Findings

Entropy is computed in terms of dynamical degrees under mild conditions.

For endomorphisms of  with degree d, h^1_{top} often equals  d, but exceptions exist.

Examples show h^1_{top} can differ from   d in some cases.

Abstract

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