Uniform hyperbolicity for random maps with positive Lyapunov exponents
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- by Yongluo Cao, Stefano Luzzatto and Isabel Rios
- Proc. Amer. Math. Soc. 136 (2008), 3591-3600
- DOI: https://doi.org/10.1090/S0002-9939-08-09347-7
- Published electronically: May 8, 2008
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Abstract:
We consider some general classes of random dynamical systems and show that a priori very weak nonuniform hyperbolicity conditions actually imply uniform hyperbolicity.References
- José F. Alves, Vítor Araújo, and Benoît Saussol, On the uniform hyperbolicity of some nonuniformly hyperbolic systems, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1303–1309. MR 1948124, DOI 10.1090/S0002-9939-02-06857-0
- Alexander Arbieto, Carlos Matheus, and Krerley Oliveira, Equilibrium states for random non-uniformly expanding maps, Nonlinearity 17 (2004), no. 2, 581–593. MR 2039060, DOI 10.1088/0951-7715/17/2/013
- Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
- Yongluo Cao, Non-zero Lyapunov exponents and uniform hyperbolicity, Nonlinearity 16 (2003), no. 4, 1473–1479. MR 1986306, DOI 10.1088/0951-7715/16/4/316
- Yongluo Cao, Stefano Luzzatto, and Isabel Rios, A minimum principle for Lyapunov exponents and a higher-dimensional version of a theorem of Mañé, Qual. Theory Dyn. Syst. 5 (2004), no. 2, 261–273. MR 2275440, DOI 10.1007/BF02972681
- Volker Matthias Gundlach and Yuri Kifer, Random hyperbolic systems, Stochastic dynamics (Bremen, 1997) Springer, New York, 1999, pp. 17–145. MR 1678467
- K. Khanin and Y. Kifer, Thermodynamic formalism for random transformations and statistical mechanics, Sinaĭ’s Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, vol. 171, Amer. Math. Soc., Providence, RI, 1996, pp. 107–140. MR 1359097, DOI 10.1090/trans2/171/10
- Yuri Kifer, Random dynamics and its applications, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 809–818. MR 1648128
- J. F. C. Kingman, Subadditive ergodic theory, Ann. Probability 1 (1973), 883–909. MR 356192, DOI 10.1214/aop/1176996798
- Pei-Dong Liu, Dynamics of random transformations: smooth ergodic theory, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1279–1319. MR 1855833, DOI 10.1017/S0143385701001614
- Pei-Dong Liu and Yang Zhao, Large deviations in random perturbations of Axiom A basic sets, J. London Math. Soc. (2) 68 (2003), no. 1, 148–164. MR 1980249, DOI 10.1112/S0024610703004290
- V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
- David Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2) 115 (1982), no. 2, 243–290. MR 647807, DOI 10.2307/1971392
- Wojciech Słomczyśki, Subadditive ergodic theorems in $C(X)$, Ital. J. Pure Appl. Math. 1 (1997), 17–28 (1998). MR 1666986
- R. Sturman and J. Stark, Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity 13 (2000), no. 1, 113–143. MR 1734626, DOI 10.1088/0951-7715/13/1/306
- Daniel H. Wagner, Survey of measurable selection theorems, SIAM J. Control Optim. 15 (1977), no. 5, 859–903. MR 486391, DOI 10.1137/0315056
Bibliographic Information
- Yongluo Cao
- Affiliation: Department of Mathematics, Suzhou University, Suzhou 215006, Jiangsu, People’s Republic of China
- MR Author ID: 343275
- Email: ylcao@suda.edu.cn
- Stefano Luzzatto
- Affiliation: Department of Mathematics, Imperial College, 180 Queen’s Gate, London SW7 2AZ, United Kingdom
- Email: Stefano.Luzzatto@imperial.ac.uk
- Isabel Rios
- Affiliation: Department of Mathematics, Universidade Federal Fluminense, Niteroi, Rio deJaneiro, Brazil
- Email: rios@mat.uff.br
- Received by editor(s): April 2, 2007
- Received by editor(s) in revised form: September 3, 2007
- Published electronically: May 8, 2008
- Additional Notes: The first author was partially supported by NSFC(10571130), NCET, and SRFDP of China and the Royal Society.
The second author was partially supported by EPSRC grant GRT0969901.
The third author was partially supported by CAPES and FAPERJ (Brazil). The authors would like thank M. Benedicks and M. Viana for their suggestions and encouragement. - Communicated by: Jane M. Hawkins
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 3591-3600
- MSC (2000): Primary 37H15
- DOI: https://doi.org/10.1090/S0002-9939-08-09347-7
- MathSciNet review: 2415043