A panorama of specification-like properties and their consequences

TL;DR

This paper reviews various specification-like properties in dynamical systems, exploring their relationships with mixing, entropy, invariant measures, and shadowing, especially within symbolic dynamics.

Contribution

It provides a comprehensive overview of the connections between specification properties and fundamental dynamical concepts, highlighting their implications in symbolic dynamics.

Findings

Specification properties relate closely to mixing and entropy.

Invariant measure structures are influenced by specification-like properties.

Connections to shadowing properties are elucidated.

Abstract

We offer an overview of the specification property, its relatives and their consequences. We examine relations between specification-like properties and such notions as: mixing, entropy, the structure of the simplex of invariant measures, and various types of the shadowing property. We pay special attention to these connections in the context of symbolic dynamics.