American Institute of Mathematical Sciences

Advanced
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 - A, 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.   Google Scholar

[2]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature,, Trudy Mat. Inst. Steklov., 90 (1967).   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, (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.  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 (1975).   Google Scholar

[7]

M. Brin, On dynamical coherence,, Ergodic Theory Dynam. Systems, 23 (2003), 395.  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.  doi: 10.3934/dcds.2008.22.89.  Google Scholar

[9]

N. Gourmelon, Adapted metric for dominated splitting,, Ergod. Theory Dyn. Syst., 27 (2007), 1839.  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, 51 (2007), 35.   Google Scholar

[11]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Math., 583 (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).   Google Scholar

[14]

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

[15]

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

[16]

S. Yu. Pilyugin, Variational shadowing,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 733.  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.  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.  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.   Google Scholar

[20]

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

[21]

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

[22]

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

[2]

D. V. Anosov, Geodesic flows on closed Riemannian manifolds of negative curvature,, Trudy Mat. Inst. Steklov., 90 (1967).   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, (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.  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 (1975).   Google Scholar

[7]

M. Brin, On dynamical coherence,, Ergodic Theory Dynam. Systems, 23 (2003), 395.  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.  doi: 10.3934/dcds.2008.22.89.  Google Scholar

[9]

N. Gourmelon, Adapted metric for dominated splitting,, Ergod. Theory Dyn. Syst., 27 (2007), 1839.  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, 51 (2007), 35.   Google Scholar

[11]

M. W. Hirsch, C. C. Pugh and M. Shub, "Invariant Manifolds,", Lecture Notes in Math., 583 (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).   Google Scholar

[14]

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

[15]

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

[16]

S. Yu. Pilyugin, Variational shadowing,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 733.  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.  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.  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.   Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[1]

Keith Burns, Amie Wilkinson. Dynamical coherence and center bunching. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 89-100. doi: 10.3934/dcds.2008.22.89
[2]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[3]

Yong Fang, Patrick Foulon, Boris Hasselblatt. Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows. Electronic Research Announcements, 2010, 17: 80-89. doi: 10.3934/era.2010.17.80
[4]

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
[5]

Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227
[6]

Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187
[7]

Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527
[8]

Raquel Ribeiro. Hyperbolicity and types of shadowing for $C^1$ generic vector fields. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2963-2982. doi: 10.3934/dcds.2014.34.2963
[9]

Michael Brin, Dmitri Burago, Sergey Ivanov. Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus. Journal of Modern Dynamics, 2009, 3 (1) : 1-11. doi: 10.3934/jmd.2009.3.1
[10]

Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641
[11]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006
[12]

Rafael Potrie. Partial hyperbolicity and foliations in $\mathbb{T}^3$. Journal of Modern Dynamics, 2015, 9: 81-121. doi: 10.3934/jmd.2015.9.81
[13]

Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477
[14]

Noriaki Kawaguchi. Topological stability and shadowing of zero-dimensional dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2743-2761. doi: 10.3934/dcds.2019115
[15]

M. Jotz. The leaf space of a multiplicative foliation. Journal of Geometric Mechanics, 2012, 4 (3) : 313-332. doi: 10.3934/jgm.2012.4.313
[16]

Marcin Mazur, Jacek Tabor, Piotr Kościelniak. Semi-hyperbolicity and hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1029-1038. doi: 10.3934/dcds.2008.20.1029
[17]

Sergei Yu. Pilyugin. Variational shadowing. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 733-737. doi: 10.3934/dcdsb.2010.14.733
[18]

Eric Bedford, Serge Cantat, Kyounghee Kim. Pseudo-automorphisms with no invariant foliation. Journal of Modern Dynamics, 2014, 8 (2) : 221-250. doi: 10.3934/jmd.2014.8.221
[19]

Marcin Mazur, Jacek Tabor. Computational hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1175-1189. doi: 10.3934/dcds.2011.29.1175
[20]

Boris Hasselblatt, Yakov Pesin, Jörg Schmeling. Pointwise hyperbolicity implies uniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2819-2827. doi: 10.3934/dcds.2014.34.2819

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences