Some relations in topological dynamics

TL;DR

This paper explores new types of relations in topological dynamics, introducing weakly distal flows and analyzing their properties, with examples like Morse-Thue and Chacon transformations.

Contribution

It introduces the concepts of strongly proximal and weakly distal relations, expanding the classification of flows in topological dynamics.

Findings

Morse-Thue substitution flows are weakly distal.

Chacon transformations are weakly distal.

Weakly distal flows form a new class in topological dynamics.

Abstract

Relations always play an important role in the study of topological dynamics. Proximal, distal and almost periodic relations are well studied in literature. We further this direction and analogously study the strongly proximal and weakly distal relations. This gives a new class of flows - the weakly distal flows. We observe that the well known Morse-Thue substitution flows and Chacon transformations are weakly distal.