The mean orbital pseudo-metric and the space of invariant measures
Jian Li, Yuanfen Xiao
TL;DR
The article develops the mean orbital pseudo-metric as a tool to study the geometry of the invariant-measure space $\mathcal{M}_T(X)$ for Polish dynamical systems. It establishes equivalent density criteria for when invariant measures generated by periodic points are dense in $\mathcal{M}_T(X)$ or its ergodic subset, via linkability/closability-type properties expressed in mean-orbital terms. A key contribution is the introduction of the asymptotic orbital average shadowing property, which guarantees that every non-empty compact connected subset of $\mathcal{M}_T(X)$ is realized by some point, strengthening the connection between orbit-asymptotics and measure realizability. The results yield concrete consequences for interval maps with zero entropy and illuminate the structure of invariant measures under various weak specification-like properties.
Abstract
We study the mean orbital pseudo-metric for Polish dynamical systems and its connections with properties of the space of invariant measures. We give equivalent conditions for when the set of invariant measures generated by periodic points is dense in the set of ergodic measures and the space of invariant measures. We also introduce the concept of asymptotic orbital average shadowing property and show that it implies that every non-empty compact connected subset of the space of invariant measures has a generic point.