Some Properties of Recurrent Sets for Endomorphisms of Topological Groups

TL;DR

This paper explores the properties of chain recurrence and shadowing in endomorphisms of topological groups, establishing that certain recurrent sets are subgroups and linking algebraic and dynamical properties.

Contribution

It generalizes chain recurrence and shadowing concepts to topological groups and proves that chain recurrent sets form subgroups, connecting algebraic and dynamical properties.

Findings

Chain recurrent points form topological subgroups.

Chain transitive components of the identity are subgroups.

Dynamical properties are preserved on quotient spaces.

Abstract

This paper studies topological definitions of chain recurrence and shadowing for continuous endomorphisms of topological groups generalizing the relevant concepts for metric spaces. It is proved that in this case the sets of chain recurrent points and chain transitive component of the identity are topological subgroups. Furthermore, it is obtained that some dynamical properties induced by the original system on quotient spaces. These results link an algebraic property to a dynamical property.