American Institute of Mathematical Sciences

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