  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 is 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 $\Omega^{\r}$-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.
