The interplay between partial specification, average shadowing, and Besicovitch completeness

TL;DR

This paper explores the relationships between partial specification, average shadowing, and measure properties in compact dynamical systems, providing new theoretical insights and examples.

Contribution

It proves that partial specification implies average shadowing and shows density of ergodic measures under surjectivity, plus an example of non-Besicovitch completeness.

Findings

Partial specification implies average shadowing.

Surjective systems with partial specification have dense ergodic measures.

An example of a non-Besicovitch complete system is provided.

Abstract

Let $(X,T)$ be a compact dynamical system. This article proves that if $(X,T)$ has the partial specification property, then it has the average shadowing property. It is also proven that if $(X,T)$ is surjective and has the partial specification property, then the set of ergodic measures of $(X,T)$ is dense in the space of its invariant measures. An example of a compact dynamical system that is not Besicovitch complete is also given.