Convergence uniforme des moyennes ergodiques des fonctions continues

TL;DR

This paper investigates the uniform convergence of ergodic averages of continuous functions, linking it to invariant functions and the system's factor structure, thus advancing the understanding of ergodic theory in continuous settings.

Contribution

It establishes conditions under which ergodic averages of continuous functions converge uniformly and relates this to the projection onto invariant functions.

Findings

Uniform convergence of ergodic averages is characterized for continuous functions.

The convergence relates to the projection on invariant functions.

Continuity of the projection is connected to the system's factor structure.

Abstract

The goal of this work is to study the space of continuous functions whose ergodic averages converge everywhere towards a continuous function. We will connect, as in the case of a metric study, the convergence of the ergodic averages and the projection of continuous functions on the subspace of invariant functions. We will see that this determines the continuity of the projection of the system onto a certain factor.