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

Partial hyperbolicity and central shadowing

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.
Citation: Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete and 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.

[2]

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

[3]

D. Bohnet and Ch. Bonatti, Partially hyperbolic diffeomorphisms with uniformly center foliation: the quotient dynamics, preprint arXiv:1210.2835.

[4]

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

[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.

[6]

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

[7]

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

[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.

[9]

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

[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.

[11]

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

[12]

Huyi Hu, Yunhua Zhou and Yujun Zhu, Quasi-Shadowing for Partially Hyperbolic Diffeomorphisms, preprint, arXiv:1210.4988.

[13]

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

[14]

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

[15]

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

[16]

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

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

S. B. Tikhomirov, Hölder shadowing on finite intervals, preprint, arXiv:1106.4053.

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.

[2]

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

[3]

D. Bohnet and Ch. Bonatti, Partially hyperbolic diffeomorphisms with uniformly center foliation: the quotient dynamics, preprint arXiv:1210.2835.

[4]

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

[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.

[6]

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

[7]

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

[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.

[9]

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

[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.

[11]

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

[12]

Huyi Hu, Yunhua Zhou and Yujun Zhu, Quasi-Shadowing for Partially Hyperbolic Diffeomorphisms, preprint, arXiv:1210.4988.

[13]

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

[14]

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

[15]

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

[16]

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

[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.

[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.

[19]

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

[20]

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

[21]

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

[22]

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

[23]

S. B. Tikhomirov, Hölder shadowing on finite intervals, preprint, arXiv:1106.4053.

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