On the closedness of ergodic measures in a characteristic class

TL;DR

This paper investigates the topological properties of ergodic measures in dynamical systems, demonstrating that within a specific metric, the set of ergodic measures in a characteristic class is closed, ensuring stability under certain convergences.

Contribution

It proves that ergodic measures in a fixed characteristic class form a closed set under a stronger topology than weak$^*$, extending previous results on measure stability.

Findings

Ergodic measures in a fixed class are closed in the $ar{ ho}$ metric.

Convergence of generic points in the Besicovitch pseudometric preserves ergodicity within the class.

The result strengthens understanding of measure stability in topological dynamics.

Abstract

We endow the set of all invariant measures of a topological dynamical system with a metric $\bar{\rho}$, which induces a topology stronger than the the weak$^*$-topology. Then, we study the closedness of ergodic measures within a characteristic class under this metric. Specifically, we show that if a sequence of generic points associated with ergodic measures from a fixed characteristic class converges in the Besicovitch pseudometric, then the limit point is generic for an ergodic measure in the same class. This implies that the set of ergodic measures belonging to a fixed characteristic class is closed in $\bar{\rho}$ (by a result of Babel, Can, Kwietniak, and Oprocha in [1]).