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July  2021, 41(7): 3093-3108. doi: 10.3934/dcds.2020399

Extensions of expansive dynamical systems

Mauricio Achigar

Departamento de Matemática y Estadística del Litoral, Centro Universitario Regional Litoral Norte, Universidad de la República, 25 de Agosto 281, Salto (50000), Uruguay

Received  October 2019 Revised  November 2020 Published  July 2021 Early access  December 2020

Fund Project: Partially supported by Agencia Nacional de Investigación e Innovación, Uruguay

We characterize and describe the extensions of expansive and Ano- sov homeomorphisms on compact spaces. As an application we obtain a stability result for extensions of Anosov systems, and show a construction that embeds any expansive system inside an expansive system having the shadowing property for the pseudo orbits of the original space.

Keywords: Semi expansive system, Semi Anosov system, Lewowicz quotient, Topological stability, Shadowing envelopes.
Mathematics Subject Classification: Primary: 37B05, 37B65; Secondary: 37B02, 37B25.
Citation: Mauricio Achigar. Extensions of expansive dynamical systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3093-3108. doi: 10.3934/dcds.2020399
References:
[1]

M. Achigar, A note on Anosov homeomorphisms, Axioms, 8 (2019), 54. doi: 10.3390/axioms8020054.  Google Scholar

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M. Achigar, A. Artigue and I. Monteverde, Expansive homeomorphisms on non-Hausdorff spaces, Topol. Appl., 207 (2016), 109-122.  doi: 10.1016/j.topol.2016.04.016.  Google Scholar

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M. Cerminara and M. Sambarino, Stable and unstable sets of $C^0$ perturbations of expansive homeomorphisms of surfaces, Nonlinearity, 12 (1999), 321-332.  doi: 10.1088/0951-7715/12/2/011.  Google Scholar

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H. B. Keynes and J. B. Robertson, Generators for topological entropy and expansiveness, Math. Systems Theory, 3 (1969), 51-59.  doi: 10.1007/BF01695625.  Google Scholar

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J. Lewowicz, Persistence in expansive systems, Ergodic Theory Dynam. Systems, 3 (1983), 567-578.  doi: 10.1017/S0143385700002157.  Google Scholar

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show all references

References:
[1]

M. Achigar, A note on Anosov homeomorphisms, Axioms, 8 (2019), 54. doi: 10.3390/axioms8020054.  Google Scholar

[2]

M. Achigar, A. Artigue and I. Monteverde, Expansive homeomorphisms on non-Hausdorff spaces, Topol. Appl., 207 (2016), 109-122.  doi: 10.1016/j.topol.2016.04.016.  Google Scholar

[3]

J. P. Aubin and H. Frankowska, Set-valued Analysis, Systems & control, Birkhäuser, 1990.  Google Scholar

[4]

B. F. Bryant, Expansive self-homeomorphisms of a compact metric space, Amer. Math. Monthly, 69 (1962), 386-391.  doi: 10.1080/00029890.1962.11989902.  Google Scholar

[5]

M. Cerminara and M. Sambarino, Stable and unstable sets of $C^0$ perturbations of expansive homeomorphisms of surfaces, Nonlinearity, 12 (1999), 321-332.  doi: 10.1088/0951-7715/12/2/011.  Google Scholar

[6]

J. L. Kelley, General Topology, D. Van Nostrand Co., 1955.  Google Scholar

[7]

H. B. Keynes and J. B. Robertson, Generators for topological entropy and expansiveness, Math. Systems Theory, 3 (1969), 51-59.  doi: 10.1007/BF01695625.  Google Scholar

[8]

J. Lewowicz, Persistence in expansive systems, Ergodic Theory Dynam. Systems, 3 (1983), 567-578.  doi: 10.1017/S0143385700002157.  Google Scholar

[9]

S. Nadler, Continuum Theory: An Introduction, Chapman & Hall/CRC Pure and Applied Mathematics, Taylor & Francis, 1992. doi: 10.1201/9781315274089.  Google Scholar

[10]

P. Walters, On the pseudo orbit tracing property and its relationship to stability, The Structure of Attractors in Dynamical Systems, Lecture Notes in Math., 668 (1978), 231-244.  doi: 10.1007/BFb0101795.  Google Scholar

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