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

TL;DR

This paper explores the relationship between exterior spaces and discrete semi-flows, analyzing limits, end points, and regions of attraction to understand their topological and dynamical properties.

Contribution

It introduces a canonical exterior space structure on discrete semi-flows and links various limit and end point concepts to dynamical features like attractors and basins.

Findings

Discrete semi-flows can be endowed with an exterior space structure.

End points decompose the region of attraction into basins.

Connections established between limits, end points, and dynamical behavior.

Abstract

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.