On the closedness of ergodic measures in a characteristic class
Sejal Babel, Martha Łącka
TL;DR
The paper investigates the closedness of ergodic invariant measures within a fixed characteristic class under the stronger $\bar{\rho}$-topology. It develops a bridge between Besicovitch quasi-genericity for points and $\bar{\rho}$-convergence of measures via joinings, and extends existing results to general compact spaces. The authors prove that for any characteristic class $\mathscr{C}$, the set of ergodic measures in $\mathscr{C}$ is closed in $\bar{\rho}$, by establishing Besicovitch-to-$\bar{\rho}$ correspondences and an ergodic-extension of BKLR19. This work preserves class-properties under limits and advances understanding of how ergodic measures behave under strong-topology convergence, with implications for dynamical applications related to orthogonality and rigidity phenomena.
Abstract
We endow the set of all invariant measures of a topological dynamical system with a metric $\barρ$, 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 $\barρ$ (by a result of Babel, Can, Kwietniak, and Oprocha in [1]).