A completion construction for continuous dynamical systems

TL;DR

This paper introduces new methods to complete continuous dynamical systems, linking topological and dynamical properties, and characterizes when these completions are homeomorphic to the original system.

Contribution

It constructs the $ ext{C}^ $- and $ ext{C}^ ext{l}$-completions of flows and explores their topological and dynamical properties, providing a framework for understanding flow completions.

Findings

Constructed canonical maps to the completions.

Identified conditions for the maps to be homeomorphisms.

Established relations between topological and dynamical properties.

Abstract

In this work we construct the $\Co^{\r}$-completion and $\Co^{\l}$-completion of a dynamical system. If $X$ is a flow, we construct canonical maps $X\to \Co^{\r}(X)$ and $X\to \Co^{\l}(X)$ and when these maps are homeomorphism we have the class of $\Co^{\r}$-complete and $\Co^{\l}$-complete flows, respectively. In this study we find out many relations between the topological properties of the completions and the dynamical properties of a given flow. In the case of a complete flow this gives interesting relations between the topological properties (separability properties, compactness, convergence of nets, etc.) and dynamical properties (periodic points, omega limits, attractors, repulsors, etc.).