Compactifications of Dynamical Systems

TL;DR

This paper develops a general theory for compactifying locally compact dynamical systems without introducing artificial recurrence, enabling better analysis of systems with non-compact state spaces.

Contribution

It introduces the concept of dynamic compactification, allowing embedding of locally compact systems into compact ones while preserving their recurrence properties.

Findings

Provides a framework for compactifying systems without artificial recurrence

Ensures points with finite orbit spans remain non-wandering after compactification

Enables analysis of non-compact dynamical systems using compact models

Abstract

While compactness is an essential assumption for many results in dynamical systems theory, for many applications the state space is only locally compact. Here we provide a general theory for compactifying such systems, i.e. embedding them as invariant open subsets of compact systems. In the process we don't want to introduce recurrence which was not there in the original system. For example if a point lies on an orbit which remains in any compact set for only a finite span of time then the point becomes non-wandering if we use the one-point compactification. Instead, we develop here the appropriate theory of dynamic compactification.