Homology, Homotopy and Applications

Volume 21 (2019)

Number 2

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

Pages: 83 – 106

DOI: https://dx.doi.org/10.4310/HHA.2019.v21.n2.a6

Authors

J.M. García Calcines (Dpto. de Matemáticas, Estad. e I.O., Universidad de La Laguna, Spain)

L.J. Hernández Paricio (Dpto. de Matemáticas y Computación, Universidad de La Rioja, Logroño, Spain)

M. Marañón Grandes (Dpto. de Matemáticas y Computación, Universidad de La Rioja, Logroño, Spain)

M.T. Rivas Rodríguez (Dpto. de Matemáticas y Computación, Universidad de La Rioja, Logroño, Spain)

Abstract

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.

Keywords

discrete semi-flow, exterior space, limit space, end point, end space, exterior discrete semi-flow

2010 Mathematics Subject Classification

54H20, 55P55, 55P57

Received 19 June 2018

Accepted 25 October 2018

Published 19 December 2018