Homology, Homotopy and Applications
Volume 21 (2019)
Regions of attraction, limits and end points of an exterior discrete semi-flow
Pages: 83 – 106
An exterior space is a topological space equipped with a distinguished quasi-filter of open subsets (closed by finite intersections) that we call externology. For an exterior space one can consider limits, bar-limits and different sets of end points (Steenrod, Čech, Brown–Grossman).
In this work we analyze relations between exterior spaces and discrete semi-flows. In order to do this we introduce the notion of exterior discrete semi-flow, which is a mixture of exterior space and discrete semi-flow. We see that any classical discrete semi-flow can be provided with the structure of an exterior discrete semi-flow by taking the quasi-filter of right-absorbing open subsets. Such a family of open subsets is used to study the relations between limits and periodic points and connections between bar-limits and omega-limits. The different notions of end points are used to decompose the region of attraction of an exterior discrete semi-flow as a disjoint union of basins of end points. We also analyze the exterior discrete semi-flow structure induced by the family of open neighborhoods of a given sub-semi-flow.
discrete semi-flow, exterior space, limit space, end point, end space, exterior discrete semi-flow
2010 Mathematics Subject Classification
54H20, 55P55, 55P57
Partially supported by Ministerio de Economía y Competitividad (grant MTM2016-78647-P) and University of La Rioja (projects: APPI16/03, EGI16/42).
Received 19 June 2018
Accepted 25 October 2018
Published 19 December 2018