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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  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

[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

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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