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July  2013, 33(7): 2901-2909. doi: 10.3934/dcds.2013.33.2901

Partial hyperbolicity and central shadowing

Sergey Kryzhevich 1, and  Sergey Tikhomirov 2,

1. 

Faculty of Mathematics and Mechanics and Chebyshev laboratory, Saint-Petersburg State University Universitetsky pr., 28, 198504, Peterhof, St. Petersburg, Russian Federation
2. 

Institut fur Mathematik, Freie Universitat Berlin, Arnimallee 3, Berlin, 14195, Germany

Received  March 2012 Revised  November 2012 Published  January 2013

We study shadowing property for a partially hyperbolic diffeomorphism $f$. It is proved that if $f$ is dynamically coherent then any pseudotrajectory can be shadowed by a pseudotrajectory with ``jumps'' along the central foliation. The proof is based on the Tikhonov-Shauder fixed point theorem.
Keywords: central foliation, Partial hyperbolicity, dynamical coherence., Lipschitz shadowing.
Mathematics Subject Classification: 37C50, 37D3.
Citation: Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901
References:
[1]

F. Abdenur and L. Diaz, Pseudo-orbit shadowing in the $C^1$ topology, Discrete Contin. Dyn. Syst., 7 (2003), 223-245.  Google Scholar

[2]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209 pp.  Google Scholar

[3]

D. Bohnet and Ch. Bonatti, Partially hyperbolic diffeomorphisms with uniformly center foliation: the quotient dynamics,, preprint , ().   Google Scholar

[4]

Ch. Bonatti, L. J. Diaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective," Springer, Berlin, 2004.  Google Scholar

[5]

Ch. Bonatti, L. Diaz and G. Turcat, There is no shadowing lemma for partially hyperbolic dynamics, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 587-592. doi: 10.1016/S0764-4442(00)00215-9.  Google Scholar

[6]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes Math., 470, Springer, Berlin, 1975.  Google Scholar

[7]

M. Brin, On dynamical coherence, Ergodic Theory Dynam. Systems, 23 (2003), 395-401. doi: 10.1017/S0143385702001499.  Google Scholar

[8]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete and Continuous Dynamical Systems, 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89.  Google Scholar

[9]

N. Gourmelon, Adapted metric for dominated splitting, Ergod. Theory Dyn. Syst., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272.  Google Scholar

[10]

F. Rodriguez-Hertz, M. A. Rodriguez-Hertz and R. Ures, A survey of partially hyperbolic dynamics, Fields Institute Communications, Partially Hyperbolic Dynamics, Laminations and Teichmuller Flow, 51 (2007), 35-88.  Google Scholar

[11]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Math., 583, Springer-Verlag, Berlin-Heidelberg, 1977.  Google Scholar

[12]

Huyi Hu, Yunhua Zhou and Yujun Zhu, Quasi-Shadowing for Partially Hyperbolic Diffeomorphisms,, preprint, ().   Google Scholar

[13]

A. Morimoto, The method of pseudo-orbit tracing and stability of dynamical systems, Sem. Note, 39 (1979), Tokyo Univ. Google Scholar

[14]

K. J. Palmer, "Shadowing in Dynamical Systems, Theory and Applications," Kluwer, Dordrecht, 2000.  Google Scholar

[15]

S. Yu. Pilyugin, "Shadowing in Dynamical Systems," Lecture Notes in Math., 1706, Springer, Berlin, 1999.  Google Scholar

[16]

S. Yu. Pilyugin, Variational shadowing, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 733-737. doi: 10.3934/dcdsb.2010.14.733.  Google Scholar

[17]

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

[18]

C. C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, revisited, J. of Modern Dynamics, 6 (2012), 79-120. doi: 10.3934/jmd.2012.6.79.  Google Scholar

[19]

C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mount. J. Math., 7 (1977), 425-437.  Google Scholar

[20]

K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms of closed manifolds, Osaka J. Math., 31 (1994), 373-386.  Google Scholar

[21]

K. Sawada, Extended f-orbits are approximated by orbits, Nagoya Math. J., 79 (1980), 33-45.  Google Scholar

[22]

J. Schauder, Der fixpunktsatz in funktionalraumen, Stud. Math., 2 (1930), 171-180. Google Scholar

[23]

S. B. Tikhomirov, Hölder shadowing on finite intervals,, preprint, ().   Google Scholar

show all references

References:
[1]

F. Abdenur and L. Diaz, Pseudo-orbit shadowing in the $C^1$ topology, Discrete Contin. Dyn. Syst., 7 (2003), 223-245.  Google Scholar

[2]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature, Trudy Mat. Inst. Steklov., 90 (1967), 209 pp.  Google Scholar

[3]

D. Bohnet and Ch. Bonatti, Partially hyperbolic diffeomorphisms with uniformly center foliation: the quotient dynamics,, preprint , ().   Google Scholar

[4]

Ch. Bonatti, L. J. Diaz and M. Viana, "Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective," Springer, Berlin, 2004.  Google Scholar

[5]

Ch. Bonatti, L. Diaz and G. Turcat, There is no shadowing lemma for partially hyperbolic dynamics, C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), 587-592. doi: 10.1016/S0764-4442(00)00215-9.  Google Scholar

[6]

R. Bowen, "Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms," Lecture Notes Math., 470, Springer, Berlin, 1975.  Google Scholar

[7]

M. Brin, On dynamical coherence, Ergodic Theory Dynam. Systems, 23 (2003), 395-401. doi: 10.1017/S0143385702001499.  Google Scholar

[8]

K. Burns and A. Wilkinson, Dynamical coherence and center bunching, Discrete and Continuous Dynamical Systems, 22 (2008), 89-100. doi: 10.3934/dcds.2008.22.89.  Google Scholar

[9]

N. Gourmelon, Adapted metric for dominated splitting, Ergod. Theory Dyn. Syst., 27 (2007), 1839-1849. doi: 10.1017/S0143385707000272.  Google Scholar

[10]

F. Rodriguez-Hertz, M. A. Rodriguez-Hertz and R. Ures, A survey of partially hyperbolic dynamics, Fields Institute Communications, Partially Hyperbolic Dynamics, Laminations and Teichmuller Flow, 51 (2007), 35-88.  Google Scholar

[11]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds," Lecture Notes in Math., 583, Springer-Verlag, Berlin-Heidelberg, 1977.  Google Scholar

[12]

Huyi Hu, Yunhua Zhou and Yujun Zhu, Quasi-Shadowing for Partially Hyperbolic Diffeomorphisms,, preprint, ().   Google Scholar

[13]

A. Morimoto, The method of pseudo-orbit tracing and stability of dynamical systems, Sem. Note, 39 (1979), Tokyo Univ. Google Scholar

[14]

K. J. Palmer, "Shadowing in Dynamical Systems, Theory and Applications," Kluwer, Dordrecht, 2000.  Google Scholar

[15]

S. Yu. Pilyugin, "Shadowing in Dynamical Systems," Lecture Notes in Math., 1706, Springer, Berlin, 1999.  Google Scholar

[16]

S. Yu. Pilyugin, Variational shadowing, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 733-737. doi: 10.3934/dcdsb.2010.14.733.  Google Scholar

[17]

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

[18]

C. C. Pugh, M. Shub and A. Wilkinson, Hölder foliations, revisited, J. of Modern Dynamics, 6 (2012), 79-120. doi: 10.3934/jmd.2012.6.79.  Google Scholar

[19]

C. Robinson, Stability theorems and hyperbolicity in dynamical systems, Rocky Mount. J. Math., 7 (1977), 425-437.  Google Scholar

[20]

K. Sakai, Pseudo orbit tracing property and strong transversality of diffeomorphisms of closed manifolds, Osaka J. Math., 31 (1994), 373-386.  Google Scholar

[21]

K. Sawada, Extended f-orbits are approximated by orbits, Nagoya Math. J., 79 (1980), 33-45.  Google Scholar

[22]

J. Schauder, Der fixpunktsatz in funktionalraumen, Stud. Math., 2 (1930), 171-180. Google Scholar

[23]

S. B. Tikhomirov, Hölder shadowing on finite intervals,, preprint, ().   Google Scholar

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