Lipschitz perturbations of expansive systems

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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

Abstract
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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 and 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.
[2]	A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Commun. Math. Phys., 126 (1989), 249-262. doi: 10.1007/BF02125125.
[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.
[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.
[5]	K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math., 27 (1990), 117-162.
[6]	E. Jeandenans, Difféomorphismes Hyperboliques Des Surfaces Et Combinatoire Des Partitions De Markov, Thesis, 1996.
[7]	J. Lewowicz, Lyapunov functions and topological stability, J. Diff. Eq., 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2.
[8]	J. Lewowicz, Persistence in expansive systems, Erg. Th. & Dyn. Sys., 3 (1983), 567-578. doi: 10.1017/S0143385700002157.
[9]	J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.), 20 (1989), 113-133. doi: 10.1007/BF02585472.
[10]	R. Mañé, Expansive diffeomorphisms, Lecture Notes in Math., Springer, Berlin, 468 (1975), 162-174.
[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.
[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.
[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.
[14]	C. Robinson, Dynamical Systems, CRC Press, $2^{nd}$ edition, 1999.
[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.
[16]	K. Takaki, Lipeomorphisms close to an Anosov diffeomorphism, Nagoya Math. J., 53 (1974), 71-82.
[17]	P. Walters, On the pseudo orbit tracing property and its relationship to stability, Lecture Notes in Math., Springer, 668 (1978), 231-244.
[18]	F. W. Wilson, Pasting diffeomorphisms of $R^n$, Illinois J. Math., 16 (1972), 222-233.

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.
[2]	A. Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Commun. Math. Phys., 126 (1989), 249-262. doi: 10.1007/BF02125125.
[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.
[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.
[5]	K. Hiraide, Expansive homeomorphisms of compact surfaces are pseudo-Anosov, Osaka J. Math., 27 (1990), 117-162.
[6]	E. Jeandenans, Difféomorphismes Hyperboliques Des Surfaces Et Combinatoire Des Partitions De Markov, Thesis, 1996.
[7]	J. Lewowicz, Lyapunov functions and topological stability, J. Diff. Eq., 38 (1980), 192-209. doi: 10.1016/0022-0396(80)90004-2.
[8]	J. Lewowicz, Persistence in expansive systems, Erg. Th. & Dyn. Sys., 3 (1983), 567-578. doi: 10.1017/S0143385700002157.
[9]	J. Lewowicz, Expansive homeomorphisms of surfaces, Bol. Soc. Brasil. Mat. (N.S.), 20 (1989), 113-133. doi: 10.1007/BF02585472.
[10]	R. Mañé, Expansive diffeomorphisms, Lecture Notes in Math., Springer, Berlin, 468 (1975), 162-174.
[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.
[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.
[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.
[14]	C. Robinson, Dynamical Systems, CRC Press, $2^{nd}$ edition, 1999.
[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.
[16]	K. Takaki, Lipeomorphisms close to an Anosov diffeomorphism, Nagoya Math. J., 53 (1974), 71-82.
[17]	P. Walters, On the pseudo orbit tracing property and its relationship to stability, Lecture Notes in Math., Springer, 668 (1978), 231-244.
[18]	F. W. Wilson, Pasting diffeomorphisms of $R^n$, Illinois J. Math., 16 (1972), 222-233.

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