A completion construction for continuous dynamical systems

Abstract

In this work we use the theory of exterior spaces to construct a $\check{C}_{0}^{\mathbf{r}}$-completion and a $\check{C}_{0}^{\mathbf{l}}$-completion of a dynamical system. If $X$ is a flow, we construct canonical maps $X\to \check{C}_{0}^{\mathbf{r}}(X)$ and $X\to \check{C}_{0}^{\mathbf{l}}(X)$ and when these maps are homeomorphisms we have the class of $\check{C}_{0}^{\mathbf{r}}$-complete and $\check{C}_{0}^{\mathbf{l}}$-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.).