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February  2014, 34(2): 869-882. doi: 10.3934/dcds.2014.34.869

Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations

Yujun Zhu 1,

1. 

College of Mathematics and Information Science, and Hebei Key Laboratory of Computational Mathematics and Applications, Hebei Normal University, Shijiazhuang, 050024, China

Received  July 2012 Revised  May 2013 Published  August 2013

In this paper, $C^0$ random perturbations of a partially hyperbolic diffeomorphism are considered. It is shown that a partially hyperbolic diffeomorphism is quasi-stable under such perturbations.
Keywords: Partial hyperbolicity, topological quasi-stability, random perturbation..
Mathematics Subject Classification: Primary: 37D20; Secondary: 37H9.
Citation: Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869
References:
[1]

L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, New York, 1998.  Google Scholar

[2]

M. Brin and Ya. Pesin, Partially hyperbolic dynamical systems, Math. USSR-Izv., 8 (1974), 177-218. doi: 10.1070/IM1974v008n01ABEH002101.  Google Scholar

[3]

A. Fathi, M. Herman and J. Yoccoz, A proof of Pesin's stable manifold theorem, in "Geometric Dynamics," Lect. Notes in Math., 1007, Springer-Verlag, Berlin-Heidelberg, (1983), 177-215. doi: 10.1007/BFb0061417.  Google Scholar

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H. Hu and Y. Zhu, Quasi-stability of partially hyperbolic diffeomorphisms,, to appear in Tran. Amer. Math. Soc., ().   Google Scholar

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Y. Kifer, "Random Perturbations of Dynamical Systems," Progress in Probability and Statistics, 16, Birkhäuser Boston, Inc., Boston, MA, 1988. doi: 10.1007/978-1-4615-8181-9.  Google Scholar

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P.-D. Liu, Dynamics of random transformations: Smooth ergodic theory, Ergo. Theo. Dyn. Syst., 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614.  Google Scholar

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P.-D. Liu, Random perturbations of Axiom A basic sets, J. Stat. Phys., 90 (1998), 467-490. doi: 10.1023/A:1023280407906.  Google Scholar

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P.-D. Liu and M. Qian, "Smooth Ergodic Theory of Random Dynamical Systems," Lect. Notes in Math., 1606, Springer-Verlag, Berlin, 1995.  Google Scholar

[11]

Q. X. Liu and P. D. Liu, Topological stability of hyperbolic sets of flows under random perturbations, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 117-127. doi: 10.3934/dcdsb.2010.13.117.  Google Scholar

[12]

Ya. Pesin, "Lectures on Partial Hyperbolicity and Stable Ergodicity," Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/003.  Google Scholar

[13]

P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78. doi: 10.1016/0040-9383(70)90051-0.  Google Scholar

[14]

Y. Zhu, J. Zhang and L. He, Shadowing and inverse shadowing for $C^1$ endomorphisms, Acta Mathematica Sinica (Engl. Ser.), 22 (2006), 1321-1328. doi: 10.1007/s10114-005-0739-6.  Google Scholar

show all references

References:
[1]

L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, New York, 1998.  Google Scholar

[2]

M. Brin and Ya. Pesin, Partially hyperbolic dynamical systems, Math. USSR-Izv., 8 (1974), 177-218. doi: 10.1070/IM1974v008n01ABEH002101.  Google Scholar

[3]

A. Fathi, M. Herman and J. Yoccoz, A proof of Pesin's stable manifold theorem, in "Geometric Dynamics," Lect. Notes in Math., 1007, Springer-Verlag, Berlin-Heidelberg, (1983), 177-215. doi: 10.1007/BFb0061417.  Google Scholar

[4]

M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Bull. Amer. Math. Soc., 76 (1970), 1015-1019. doi: 10.1090/S0002-9904-1970-12537-X.  Google Scholar

[5]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lect. Notes in Math., 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[6]

H. Hu and Y. Zhu, Quasi-stability of partially hyperbolic diffeomorphisms,, to appear in Tran. Amer. Math. Soc., ().   Google Scholar

[7]

Y. Kifer, "Random Perturbations of Dynamical Systems," Progress in Probability and Statistics, 16, Birkhäuser Boston, Inc., Boston, MA, 1988. doi: 10.1007/978-1-4615-8181-9.  Google Scholar

[8]

P.-D. Liu, Dynamics of random transformations: Smooth ergodic theory, Ergo. Theo. Dyn. Syst., 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614.  Google Scholar

[9]

P.-D. Liu, Random perturbations of Axiom A basic sets, J. Stat. Phys., 90 (1998), 467-490. doi: 10.1023/A:1023280407906.  Google Scholar

[10]

P.-D. Liu and M. Qian, "Smooth Ergodic Theory of Random Dynamical Systems," Lect. Notes in Math., 1606, Springer-Verlag, Berlin, 1995.  Google Scholar

[11]

Q. X. Liu and P. D. Liu, Topological stability of hyperbolic sets of flows under random perturbations, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 117-127. doi: 10.3934/dcdsb.2010.13.117.  Google Scholar

[12]

Ya. Pesin, "Lectures on Partial Hyperbolicity and Stable Ergodicity," Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/003.  Google Scholar

[13]

P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78. doi: 10.1016/0040-9383(70)90051-0.  Google Scholar

[14]

Y. Zhu, J. Zhang and L. He, Shadowing and inverse shadowing for $C^1$ endomorphisms, Acta Mathematica Sinica (Engl. Ser.), 22 (2006), 1321-1328. doi: 10.1007/s10114-005-0739-6.  Google Scholar

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