Limit and end functors of dynamical systems via exterior spaces

Abstract

In this paper we analyze some applications of the category of exterior spaces to the study of dynamical systems (flows). We study the notion of an absorbing open subset of a dynamical system; i.e., an open subset that contains the ``future part'' of all the trajectories. The family of all absorbing open subsets is a quasi-filter which gives the structure of an exterior space to the flow. The limit space and end space of an exterior space are used to construct the limit spaces and end spaces of a dynamical system. On the one hand, for a dynamical system two limits spaces $L^{\r}(X)$ and $\bar L^{\r}(X)$ are constructed and their relations with the subflows of periodic, Poisson stable points and $\Omega $-limits of $X$ are analyzed. On the other hand, different end spaces are also associated to a dynamical system having the property that any positive semi-trajectory has an end point in these end spaces. This type of construction permits us to consider the subflow containing all trajectories finishing at an end point $a$. When $a$ runs over the set of all end points, we have an induced decomposition of a dynamical system as a disjoint union of stable (at infinity) subflows.