Random dynamics on real and complex projective surfaces

TL;DR

This paper explores the behavior of random automorphisms on real and complex projective surfaces, focusing on classifying stationary measures and establishing conditions for their invariance, uniqueness, and smoothness.

Contribution

It introduces a framework for understanding stationary measures in the context of random automorphisms on Kähler surfaces, combining tools from geometry, dynamics, and matrix theory.

Findings

Stationary measures are often invariant under random automorphisms.

Criteria for the uniqueness and smoothness of invariant measures are established.

The study connects complex geometry, hyperbolicity, and random matrix products in surface dynamics.

Abstract

We initiate the study of random iteration of automorphisms of real and complex projective surfaces, or more generally compact K{\"a}hler surfaces, focusing on the fundamental problem of classification of stationary measures. We show that, in a number of cases, such stationary measures are invariant, and provide criteria for uniqueness, smoothness and rigidity of invariant probability measures. This involves a variety of tools from complex and algebraic geometry, random products of matrices, non-uniform hyperbolicity, as well as recent results of Brown and Rodriguez Hertz on random iteration of surface diffeomorphisms.