When all points are generic for ergodic measures

TL;DR

This paper explores the relationships between various properties of topological dynamical systems, including generic points, measure continuity, ergodic decompositions, and uniform convergence of Cesaro means.

Contribution

It establishes new connections between these properties, providing a unified framework for understanding ergodic measures and system decompositions.

Findings

Points are generic for ergodic measures under certain conditions

Continuity of measure assignment maps is linked to system decompositions

Cesaro means of continuous functions converge uniformly in these systems

Abstract

We establish connections between several properties of topological dynamical systems, such as: - every point is generic for an ergodic measure, - the map sending points to the measures they generate is continuous, - the system splits into uniquely (alternatively, strictly) ergodic subsystems, - the map sending ergodic measures to their topological supports is continuous, - the Cesaro means of every continuous function converge uniformly.