On specification and measure expansiveness

TL;DR

This paper explores the relationships between various specification and expansiveness properties in dynamical systems, introduces strong measure expansiveness, and proves a spectral decomposition theorem for such systems with shadowing.

Contribution

It introduces the concept of strong measure expansiveness, clarifies relations between local specification and measure expansiveness, and establishes a spectral decomposition theorem.

Findings

Strong measure expansiveness is a distinct property with examples.

Non-expansive systems can be strongly measure expansive.

Spectral decomposition holds for strong measure expansive systems with shadowing.

Abstract

We relate the local specification and periodic shadowing properties. We also clarify the relation between local weak specification and local specification if the system is measure expansive. The notion of strong measure expansiveness is introduced, and an example of a non-expansive systems with the strong measure expansive property is given. Moreover, we find a family of examples with the $N$-expansive property, which are not strong measure expansive. We finally show a spectral decomposition theorem for strong measure expansive dynamical systems with shadowing.