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 & 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]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409

[2]

Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021022

[3]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393

[4]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91

[5]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

[6]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[7]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[8]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[9]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (11)

Other articles
by authors

[Back to Top]