Closed orbits in quotient systems

TL;DR

This paper investigates how closed orbits behave in quotient dynamical systems formed by group actions, establishing bounds and possible growth rates based on group properties.

Contribution

It provides bounds and realizability results for the growth of closed orbits in quotient systems under finite group actions.

Findings

Derived upper and lower bounds for orbit growth in quotient systems

Identified three phenomena influencing orbit behavior

Showed that any growth rate within bounds can be achieved

Abstract

We study the relationship between pairs of topological dynamical systems $ (X,T) $ and $ (X',T') $, where $ (X',T') $ is the quotient of $ (X,T) $ under the action of a finite group $ G $. We describe three phenomena concerning the behaviour of closed orbits in the quotient system, and the constraints given by these phenomena. We find upper and lower bounds for the extremal behaviour of closed orbits in the quotient system in terms of properties of $ G $ and show that any growth rate in between these bounds can be achieved.