Propri\'et\'es ergodiques des applications rationnelles

TL;DR

This survey explores the construction and properties of canonical invariant measures for meromorphic endomorphisms on compact Kähler manifolds, emphasizing ergodic and hyperbolic features under certain conditions.

Contribution

It provides a comprehensive overview of methods to construct and analyze invariant measures with ergodic properties for meromorphic maps on Kähler manifolds.

Findings

Existence of canonical invariant measures with maximal entropy

Conditions ensuring ergodicity and hyperbolicity

Connections between numerical conditions and measure properties

Abstract

This is a survey article with focus on the following problem. Given $f:X \to X$ a meromorphic endomorphism of some compact K\"ahler manifold $X$, construct and study - under natural numerical conditions - a canonical invariant probability measure with remarkable ergodic properties (mixing, hyperbolicity, maximal entropy, etc).