American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media
  • DCDS Home
  • This Issue
  • Next Article
    Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions
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

[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]

Kyungwoo Song, Yuxi Zheng. Semi-hyperbolic patches of solutions of the pressure gradient system. Discrete & Continuous Dynamical Systems, 2009, 24 (4) : 1365-1380. doi: 10.3934/dcds.2009.24.1365
[2]

Michael Frankel, Victor Roytburd, Gregory I. Sivashinsky. Dissipativity for a semi-linearized system modeling cellular flames. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 83-99. doi: 10.3934/dcdss.2011.4.83
[3]

Noriaki Kawaguchi. Topological stability and shadowing of zero-dimensional dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2743-2761. doi: 10.3934/dcds.2019115
[4]

Elisabetta Carlini, Francisco J. Silva. A semi-Lagrangian scheme for a degenerate second order mean field game system. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4269-4292. doi: 10.3934/dcds.2015.35.4269
[5]

Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267
[6]

Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051
[7]

Messoud Efendiev, Etsushi Nakaguchi, Wolfgang L. Wendland. Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system. Conference Publications, 2007, 2007 (Special) : 334-343. doi: 10.3934/proc.2007.2007.334
[8]

Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1525-1539. doi: 10.3934/cpaa.2014.13.1525
[9]

Woochul Jung, Ngocthach Nguyen, Yinong Yang. Spectral decomposition for rescaling expansive flows with rescaled shadowing. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2267-2283. doi: 10.3934/dcds.2020113
[10]

Nguyen Thieu Huy, Vu Thi Ngoc Ha, Pham Truong Xuan. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2103-2116. doi: 10.3934/cpaa.2016029
[11]

Audric Drogoul, Gilles Aubert. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Problems & Imaging, 2016, 10 (1) : 51-86. doi: 10.3934/ipi.2016.10.51
[12]

P.E. Kloeden, Victor S. Kozyakin. The perturbation of attractors of skew-product flows with a shadowing driving system. Discrete & Continuous Dynamical Systems, 2001, 7 (4) : 883-893. doi: 10.3934/dcds.2001.7.883
[13]

Artur O. Lopes, Vladimir A. Rosas, Rafael O. Ruggiero. Cohomology and subcohomology problems for expansive, non Anosov geodesic flows. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 403-422. doi: 10.3934/dcds.2007.17.403
[14]

Enoch Humberto Apaza Calla, Bulmer Mejia Garcia, Carlos Arnoldo Morales Rojas. Topological properties of sectional-Anosov flows. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4735-4741. doi: 10.3934/dcds.2015.35.4735
[15]

Marcin Mazur, Jacek Tabor, Piotr Kościelniak. Semi-hyperbolicity and hyperbolicity. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 1029-1038. doi: 10.3934/dcds.2008.20.1029
[16]

Alfonso Artigue. Discrete and continuous topological dynamics: Fields of cross sections and expansive flows. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5911-5927. doi: 10.3934/dcds.2016059
[17]

Jaume Llibre, Jesús S. Pérez del Río, J. Angel Rodríguez. Structural stability of planar semi-homogeneous polynomial vector fields applications to critical points and to infinity. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 809-828. doi: 10.3934/dcds.2000.6.809
[18]

Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661
[19]

Xiaodong Fan, Tian Qin. Stability analysis for generalized semi-infinite optimization problems under functional perturbations. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1221-1233. doi: 10.3934/jimo.2018201
[20]

Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (160)
  • HTML views (180)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy