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

Lipschitz perturbations of expansive systems

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.
Citation: Alfonso Artigue. Lipschitz perturbations of expansive systems. Discrete & Continuous Dynamical Systems - A, 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). doi: 10.4064/dm472-0-1.

[2]

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

[3]

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

[4]

J. Harrison, Unsmoothable diffeomorphisms on higher dimensional manifolds,, Proc. Amer. Math. Soc., 73 (1979), 249. 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.

[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. doi: 10.1016/0022-0396(80)90004-2.

[8]

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

[9]

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

[10]

R. Mañé, Expansive diffeomorphisms,, Lecture Notes in Math., 468 (1975), 162.

[11]

H. Masur and J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms,, Comment. Math. Helvetici, 68 (1993), 289. 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. doi: 10.3934/dcds.2003.9.287.

[13]

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

[14]

C. Robinson, Dynamical Systems,, CRC Press, (1999).

[15]

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

[16]

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

[17]

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

[18]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

J. Harrison, Unsmoothable diffeomorphisms on higher dimensional manifolds,, Proc. Amer. Math. Soc., 73 (1979), 249. 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.

[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. doi: 10.1016/0022-0396(80)90004-2.

[8]

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

[9]

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

[10]

R. Mañé, Expansive diffeomorphisms,, Lecture Notes in Math., 468 (1975), 162.

[11]

H. Masur and J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms,, Comment. Math. Helvetici, 68 (1993), 289. 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. doi: 10.3934/dcds.2003.9.287.

[13]

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

[14]

C. Robinson, Dynamical Systems,, CRC Press, (1999).

[15]

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

[16]

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

[17]

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

[18]

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

[1]

Davor Dragičević. Admissibility, a general type of Lipschitz shadowing and structural stability. Communications on Pure & Applied Analysis, 2015, 14 (3) : 861-880. doi: 10.3934/cpaa.2015.14.861

[2]

Jorge Groisman. Expansive homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 213-239. doi: 10.3934/dcds.2011.29.213

[3]

João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837

[4]

Jorge Groisman. Expansive and fixed point free homeomorphisms of the plane. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1709-1721. doi: 10.3934/dcds.2012.32.1709

[5]

Alfonso Artigue. Anomalous cw-expansive surface homeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3511-3518. doi: 10.3934/dcds.2016.36.3511

[6]

Keonhee Lee, Kazumine Moriyasu, Kazuhiro Sakai. $C^1$-stable shadowing diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 683-697. doi: 10.3934/dcds.2008.22.683

[7]

Kazuhiro Sakai. The oe-property of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 581-591. doi: 10.3934/dcds.1998.4.581

[8]

Fang Zhang, Yunhua Zhou. On the limit quasi-shadowing property. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123

[9]

Alfonso Artigue. Robustly N-expansive surface diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2367-2376. doi: 10.3934/dcds.2016.36.2367

[10]

Baojun Bian, Pengfei Guan. A structural condition for microscopic convexity principle. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 789-807. doi: 10.3934/dcds.2010.28.789

[11]

Boris Hasselblatt and Amie Wilkinson. Prevalence of non-Lipschitz Anosov foliations. Electronic Research Announcements, 1997, 3: 93-98.

[12]

Patrick Foulon, Boris Hasselblatt. Lipschitz continuous invariant forms for algebraic Anosov systems. Journal of Modern Dynamics, 2010, 4 (3) : 571-584. doi: 10.3934/jmd.2010.4.571

[13]

Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037

[14]

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

[15]

Christian Bonatti, Nancy Guelman. Axiom A diffeomorphisms derived from Anosov flows. Journal of Modern Dynamics, 2010, 4 (1) : 1-63. doi: 10.3934/jmd.2010.4.1

[16]

Maria Carvalho. First homoclinic tangencies in the boundary of Anosov diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 765-782. doi: 10.3934/dcds.1998.4.765

[17]

Jean Lerbet, Noël Challamel, François Nicot, Félix Darve. Kinematical structural stability. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 529-536. doi: 10.3934/dcdss.2016010

[18]

Mikhail Krastanov, Michael Malisoff, Peter Wolenski. On the strong invariance property for non-Lipschitz dynamics. Communications on Pure & Applied Analysis, 2006, 5 (1) : 107-124. doi: 10.3934/cpaa.2006.5.107

[19]

Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231

[20]

Lin Wang, Yujun Zhu. Center specification property and entropy for partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 469-479. doi: 10.3934/dcds.2016.36.469

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]