American Institute of Mathematical Sciences

Advanced
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 Anosov 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 - A, 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

[1]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849
[2]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935
[3]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228
[4]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203
[5]

Eduardo Casas, Christian Clason, Arnd Rösch. Preface special issue on system modeling and optimization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021008
[6]

Dugan Nina, Ademir Fernando Pazoto, Lionel Rosier. Controllability of a 1-D tank containing a fluid modeled by a Boussinesq system. Evolution Equations & Control Theory, 2013, 2 (2) : 379-402. doi: 10.3934/eect.2013.2.379
[7]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267
[8]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393
[9]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[10]

Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197
[11]

Manoel J. Dos Santos, Baowei Feng, Dilberto S. Almeida Júnior, Mauro L. Santos. Global and exponential attractors for a nonlinear porous elastic system with delay term. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2805-2828. doi: 10.3934/dcdsb.2020206
[12]

Eric Babson and Dmitry N. Kozlov. Topological obstructions to graph colorings. Electronic Research Announcements, 2003, 9: 61-68.

[13]

Antonio Rieser. A topological approach to spectral clustering. Foundations of Data Science, 2021  doi: 10.3934/fods.2021005
[14]

Zaihong Wang, Jin Li, Tiantian Ma. An erratum note on the paper: Positive periodic solution for Brillouin electron beam focusing system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1995-1997. doi: 10.3934/dcdsb.2013.18.1995
[15]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81
[16]

Carlos Fresneda-Portillo, Sergey E. Mikhailov. Analysis of Boundary-Domain Integral Equations to the mixed BVP for a compressible stokes system with variable viscosity. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3059-3088. doi: 10.3934/cpaa.2019137
[17]

Hong Yi, Chunlai Mu, Guangyu Xu, Pan Dai. A blow-up result for the chemotaxis system with nonlinear signal production and logistic source. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2537-2559. doi: 10.3934/dcdsb.2020194
[18]

Ying Yang. Global classical solutions to two-dimensional chemotaxis-shallow water system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2625-2643. doi: 10.3934/dcdsb.2020198
[19]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447
[20]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (37)
  • HTML views (96)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences