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Shadowing for Local Homeomorphisms, with Applications to Edge Shift Spaces of Infinite Graphs

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Abstract

In this paper, we develop the basic theory of the shadowing property for local homeomorphisms of metric locally compact spaces, with a focus on applications to edge shift spaces connected with C*-algebra theory. For the local homeomorphism (the Deaconu–Renault system) associated with a directed graph, we completely characterize the shadowing property in terms of conditions on sets of paths. Using these results, we single out classes of graphs for which the associated system presents the shadowing property, fully characterize the shadowing property for systems associated with certain graphs, and show that the system associated with the rose of infinite petals presents the shadowing property and that the Renewal shift system does not present the shadowing property.

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Funding

D. Gonçalves was partially supported by Capes-PrInt, Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)—Brazil, and Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina (FAPESC).

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, Brazil

    Daniel Gonçalves

  2. Instituto Federal do Rio Grande do Sul, 92412-240, Canoas, RS, Brazil

    Bruno B. Uggioni

Authors
  1. Daniel Gonçalves
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  2. Bruno B. Uggioni
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BBU and DG wrote de the main manuscript text and reviwed the manuscript.

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Correspondence to Daniel Gonçalves.

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Gonçalves, D., Uggioni, B.B. Shadowing for Local Homeomorphisms, with Applications to Edge Shift Spaces of Infinite Graphs. J Dyn Diff Equat (2024). https://doi.org/10.1007/s10884-023-10342-7

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  • DOI: https://doi.org/10.1007/s10884-023-10342-7

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