Types of mixings and transitivities in topological dynamics

TL;DR

This paper explores the classification of topological dynamical systems based on mixing and transitivity properties, extending combinatorial methods from measure-preserving systems to topological dynamics.

Contribution

It introduces new classes of topological dynamical systems by adapting combinatorial classifications from measure-preserving systems, highlighting differences and similarities.

Findings

New classes of topological dynamical systems are proposed.

Differences between measure-preserving and topological dynamics are discussed.

The paper extends combinatorial classification methods to topological settings.

Abstract

Using the combinatorial properties of subsets of integers, a classification of metric dynamical systems was given in [V. Bergelson and T. Downarowicz, Large sets of integers and hierarchy of mixing properties of measure-preserving systems, Colloquium Mathematicum, 110(1), (2008), 117-150]. As a result, some new families emerged. Here, their counterparts in topological dynamics has been considered. The differences will be discussed and new classes of systems in topological dynamics will be introduced.