Measure Expansivity and Specification for Pointwise Dynamics

TL;DR

This paper introduces pointwise measure expansivity for bi-measurable maps, explores its properties, and investigates the relationships between mixing, specification, chaos, and entropy at specific points in dynamical systems.

Contribution

It defines a weaker form of measure expansivity, extends many properties of measure expansive systems to pointwise systems, and links mixing, specification, chaos, and entropy at points.

Findings

Pointwise measure expansivity is weaker than measure expansivity.

Many measure expansive results hold for pointwise systems.

Existence of two distinct specification points implies positive Bowen entropy.

Abstract

We introduce pointwise measure expansivity for bi-measurable maps. We show through examples that this notion is weaker than measure expansivity. In spite of this fact, we show that many results for measure expansive systems hold true for pointwise systems as well. Then, we study the concept of mixing, specification and chaos at a point in the phase space of a continuous map. We show that mixing at a shadowable point is not sufficient for it to be a specification point, but mixing of the map force a shadowable point to be a specification point. We prove that periodic specification points are Devaney chaotic point. Finally, we show that existence of two distinct specification points is sufficient for a map to have positive Bowen entropy.