Properties of invariant measures in dynamical systems with the shadowing property

TL;DR

This paper studies invariant measures in dynamical systems with the shadowing property, showing they can be approximated by ergodic measures supported on odometers, and establishing density and genericity results related to entropy.

Contribution

It introduces a method to approximate invariant measures by ergodic measures supported on odometers and their extensions, revealing new density and genericity properties.

Findings

Ergodic measures supported on odometers are dense in invariant measures for systems with shadowing.

Ergodic measures are generic in the space of invariant measures.

Density of ergodic measures with specified entropy ranges under upper semi-continuity.

Abstract

For dynamical systems with the shadowing property, we provide a method of approximation of invariant measures by ergodic measures supported on odometers and their almost 1-1 extensions. For a topologically transitive system with the shadowing property, we show that ergodic measures supported on odometers are dense in the space of invariant measures, and then ergodic measures are generic in the space of invariant measures. We also show that for every $c\geq 0$ and $\varepsilon>0$ the collection of ergodic measures (supported on almost 1-1 extensions of odometers) with entropy between $c$ and $c + \varepsilon$ is dense in the space of invariant measures with entropy at least $c$. Moreover, if in addition the entropy function is upper semi-continuous, then for every $c\geq 0$ ergodic measures with entropy $c$ are generic in the space of invariant measures with entropy at least $c$.