Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations

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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

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

Abstract
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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 and 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.
[2]	M. Brin and Ya. Pesin, Partially hyperbolic dynamical systems, Math. USSR-Izv., 8 (1974), 177-218. doi: 10.1070/IM1974v008n01ABEH002101.
[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.
[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.
[5]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lect. Notes in Math., 583, Springer-Verlag, Berlin-New York, 1977.
[6]	H. Hu and Y. Zhu, Quasi-stability of partially hyperbolic diffeomorphisms,, to appear in Tran. Amer. Math. Soc., ().
[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.
[8]	P.-D. Liu, Dynamics of random transformations: Smooth ergodic theory, Ergo. Theo. Dyn. Syst., 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614.
[9]	P.-D. Liu, Random perturbations of Axiom A basic sets, J. Stat. Phys., 90 (1998), 467-490. doi: 10.1023/A:1023280407906.
[10]	P.-D. Liu and M. Qian, "Smooth Ergodic Theory of Random Dynamical Systems," Lect. Notes in Math., 1606, Springer-Verlag, Berlin, 1995.
[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.
[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.
[13]	P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78. doi: 10.1016/0040-9383(70)90051-0.
[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.

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References:

[1]	L. Arnold, "Random Dynamical Systems," Springer Monographs in Mathematics, Springer-Verlag, New York, 1998.
[2]	M. Brin and Ya. Pesin, Partially hyperbolic dynamical systems, Math. USSR-Izv., 8 (1974), 177-218. doi: 10.1070/IM1974v008n01ABEH002101.
[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.
[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.
[5]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds," Lect. Notes in Math., 583, Springer-Verlag, Berlin-New York, 1977.
[6]	H. Hu and Y. Zhu, Quasi-stability of partially hyperbolic diffeomorphisms,, to appear in Tran. Amer. Math. Soc., ().
[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.
[8]	P.-D. Liu, Dynamics of random transformations: Smooth ergodic theory, Ergo. Theo. Dyn. Syst., 21 (2001), 1279-1319. doi: 10.1017/S0143385701001614.
[9]	P.-D. Liu, Random perturbations of Axiom A basic sets, J. Stat. Phys., 90 (1998), 467-490. doi: 10.1023/A:1023280407906.
[10]	P.-D. Liu and M. Qian, "Smooth Ergodic Theory of Random Dynamical Systems," Lect. Notes in Math., 1606, Springer-Verlag, Berlin, 1995.
[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.
[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.
[13]	P. Walters, Anosov diffeomorphisms are topologically stable, Topology, 9 (1970), 71-78. doi: 10.1016/0040-9383(70)90051-0.
[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.

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