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May  2015, 35(5): 1829-1841. doi: 10.3934/dcds.2015.35.1829

Lipschitz perturbations of expansive systems

Alfonso Artigue 1,

1. 

Departamento de Matemática y Estadística del Litoral, Universidad de la República, Gral. Rivera 1350, Salto

Received  May 2014 Revised  September 2014 Published  December 2014

We extend some known results from smooth dynamical systems to the category of Lipschitz homeomorphisms of compact metric spaces. We consider dynamical properties as robust expansiveness and structural stability allowing Lipschitz perturbations with respect to a hyperbolic metric. We also study the relationship between Lipschitz topologies and the $C^1$ topology on smooth manifolds.
Keywords: shadowing property, Expansive homeomorphisms, Anosov diffeomorphisms., structural stability, Lipschitz condition.
Mathematics Subject Classification: Primary: 37B05; Secondary: 37C20, 37E3.
Citation: Alfonso Artigue. Lipschitz perturbations of expansive systems. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1829-1841. doi: 10.3934/dcds.2015.35.1829
References:
[1]

N. H. Bingham and A. J. Ostaszewski, Normed versus topological groups: Dichotomy and duality, Dissertationes Mathematicae, 472 (2010), 138pp. doi: 10.4064/dm472-0-1.  Google Scholar

[2]

A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Commun. Math. Phys., 126 (1989), 249-262. doi: 10.1007/BF02125125.  Google Scholar

[3]

K. Fukui and T. Nakamura, A topological property of Lipschitz mappings, Topology Appl., 148 (2005), 143-152. doi: 10.1016/j.topol.2004.08.005.  Google Scholar

[4]

J. Harrison, Unsmoothable diffeomorphisms on higher dimensional manifolds, Proc. Amer. Math. Soc., 73 (1979), 249-255. doi: 10.1090/S0002-9939-1979-0516473-9.  Google Scholar

[5]

K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math., 27 (1990), 117-162.  Google Scholar

[6]

E. Jeandenans, Difféomorphismes Hyperboliques Des Surfaces Et Combinatoire Des Partitions De Markov, Thesis, 1996. Google Scholar

[7]

J. Lewowicz, Lyapunov functions and topological stability, J. Diff. Eq., 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2.  Google Scholar

[8]

J. Lewowicz, Persistence in expansive systems, Erg. Th. & Dyn. Sys., 3 (1983), 567-578. doi: 10.1017/S0143385700002157.  Google Scholar

[9]

J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.), 20 (1989), 113-133. doi: 10.1007/BF02585472.  Google Scholar

[10]

R. Mañé, Expansive diffeomorphisms, Lecture Notes in Math., Springer, Berlin, 468 (1975), 162-174.  Google Scholar

[11]

H. Masur and J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helvetici, 68 (1993), 289-307. doi: 10.1007/BF02565820.  Google Scholar

[12]

S. Yu. Pilyugin, A. A. Rodionova and K. Sakai, Orbital and weak shadowing properties, Discrete and continuous dynamical systems, 9 (2003), 287-308. doi: 10.3934/dcds.2003.9.287.  Google Scholar

[13]

S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. doi: 10.1088/0951-7715/23/10/009.  Google Scholar

[14]

C. Robinson, Dynamical Systems, CRC Press, $2^{nd}$ edition, 1999.  Google Scholar

[15]

K. Sakai and R. Y. Wong, Conjugating homeomorphisms to uniform homeomorphisms, Trans. Amer. Math. Soc., 311 (1989), 337-356. doi: 10.1090/S0002-9947-1989-0974780-0.  Google Scholar

[16]

K. Takaki, Lipeomorphisms close to an Anosov diffeomorphism, Nagoya Math. J., 53 (1974), 71-82.  Google Scholar

[17]

P. Walters, On the pseudo orbit tracing property and its relationship to stability, Lecture Notes in Math., Springer, 668 (1978), 231-244.  Google Scholar

[18]

F. W. Wilson, Pasting diffeomorphisms of $R^n$, Illinois J. Math., 16 (1972), 222-233.  Google Scholar

show all references

References:
[1]

N. H. Bingham and A. J. Ostaszewski, Normed versus topological groups: Dichotomy and duality, Dissertationes Mathematicae, 472 (2010), 138pp. doi: 10.4064/dm472-0-1.  Google Scholar

[2]

A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Commun. Math. Phys., 126 (1989), 249-262. doi: 10.1007/BF02125125.  Google Scholar

[3]

K. Fukui and T. Nakamura, A topological property of Lipschitz mappings, Topology Appl., 148 (2005), 143-152. doi: 10.1016/j.topol.2004.08.005.  Google Scholar

[4]

J. Harrison, Unsmoothable diffeomorphisms on higher dimensional manifolds, Proc. Amer. Math. Soc., 73 (1979), 249-255. doi: 10.1090/S0002-9939-1979-0516473-9.  Google Scholar

[5]

K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math., 27 (1990), 117-162.  Google Scholar

[6]

E. Jeandenans, Difféomorphismes Hyperboliques Des Surfaces Et Combinatoire Des Partitions De Markov, Thesis, 1996. Google Scholar

[7]

J. Lewowicz, Lyapunov functions and topological stability, J. Diff. Eq., 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2.  Google Scholar

[8]

J. Lewowicz, Persistence in expansive systems, Erg. Th. & Dyn. Sys., 3 (1983), 567-578. doi: 10.1017/S0143385700002157.  Google Scholar

[9]

J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.), 20 (1989), 113-133. doi: 10.1007/BF02585472.  Google Scholar

[10]

R. Mañé, Expansive diffeomorphisms, Lecture Notes in Math., Springer, Berlin, 468 (1975), 162-174.  Google Scholar

[11]

H. Masur and J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helvetici, 68 (1993), 289-307. doi: 10.1007/BF02565820.  Google Scholar

[12]

S. Yu. Pilyugin, A. A. Rodionova and K. Sakai, Orbital and weak shadowing properties, Discrete and continuous dynamical systems, 9 (2003), 287-308. doi: 10.3934/dcds.2003.9.287.  Google Scholar

[13]

S. Yu. Pilyugin and S. B. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. doi: 10.1088/0951-7715/23/10/009.  Google Scholar

[14]

C. Robinson, Dynamical Systems, CRC Press, $2^{nd}$ edition, 1999.  Google Scholar

[15]

K. Sakai and R. Y. Wong, Conjugating homeomorphisms to uniform homeomorphisms, Trans. Amer. Math. Soc., 311 (1989), 337-356. doi: 10.1090/S0002-9947-1989-0974780-0.  Google Scholar

[16]

K. Takaki, Lipeomorphisms close to an Anosov diffeomorphism, Nagoya Math. J., 53 (1974), 71-82.  Google Scholar

[17]

P. Walters, On the pseudo orbit tracing property and its relationship to stability, Lecture Notes in Math., Springer, 668 (1978), 231-244.  Google Scholar

[18]

F. W. Wilson, Pasting diffeomorphisms of $R^n$, Illinois J. Math., 16 (1972), 222-233.  Google Scholar

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