Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory

TL;DR

This paper investigates the detailed dynamical properties of an ergodic equilibrium measure for meromorphic maps with small topological degree on complex surfaces, including bounds on Lyapunov exponents, entropy maximization, and equidistribution of saddle points.

Contribution

It provides a comprehensive analysis of the invariant measure's properties, extending known results to non-invertible maps with small topological degree.

Findings

Optimal bounds for Lyapunov exponents

Proved the measure has maximal entropy

Saddle points are equidistributed towards the measure

Abstract

We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the invertible case and is, to our knowledge, one among not very many instances in which a natural invariant measure for a non-invertible dynamical system is well-understood.