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 $X\to
\check{C}_{0}^\mathbf{lr(X)$ 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.).