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Regions of attraction, limits and end points of an exterior discrete semi-flow

An exterior space is a topological space provided with a quasi-filter of open subsets (closed by finite intersections). In this work, we analyze some relations between the notion of an exterior space and the notion of a discrete semi-flow. On the one hand, for an exterior space, one can consider limits, bar-limits and different sets of end points (Steenrod, \v Cech, Brown-Grossman). On the other hand, for a discrete semi-flow one can analyze fixed points, periodic points, omega-limits, et cetera. In this paper, we see that a discrete semi-flow can be provided with the canonical structure of an exterior space given by the family of right-absorbing open subsets that can be used to study the relation between limits and periodic points and connections between bar-limits and omega-limits. The different notions of end point can be used to decompose the region of attraction of an exterior discrete semi-flow as a disjoint union of basins of end points.