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Uniform hyperbolicity for random maps with positive Lyapunov exponents


Authors: Yongluo Cao, Stefano Luzzatto and Isabel Rios
Journal: Proc. Amer. Math. Soc. 136 (2008), 3591-3600
MSC (2000): Primary 37H15
DOI: https://doi.org/10.1090/S0002-9939-08-09347-7
Published electronically: May 8, 2008
MathSciNet review: 2415043
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider some general classes of random dynamical systems and show that a priori very weak nonuniform hyperbolicity conditions actually imply uniform hyperbolicity.


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  • 1. Alves, J. F., Uniform hyperbolicity of some nonuniformly hyperbolic systems, Proc. Amer. Math. Soc., 131(4) (2003) 1303-1309. MR 1948124 (2003k:37046)
  • 2. Arbieto, A.; Matheus, C.; Oliveira, K., Equilibrium states for random non-uniformly expanding maps, Nonlinearity, 17 (2004) 581-593. MR 2039060 (2005c:37010)
  • 3. Arnold, L., Random dynamical systems (1998) Springer-Verlag. MR 1723992 (2000m:37087)
  • 4. Cao, Y., Nonzero Lyapunov exponents and uniform hyperbolicity, Nonlinearity, 16 (2003) 1473-1479. MR 1986306 (2005g:37061)
  • 5. Cao, Y.; Rios, I.; Luzzatto, S., A minimum principle for Lyapunov exponents and a higher-dimensional version of a theorem of Mané, Qualitative Theory of Dynamical Systems, 5 (2004) 261-273. MR 2275440 (2007k:37031)
  • 6. Gundlach,V. M.; Kifer,Y., Random hyperbolic systems, Stochastic dynamics (Bremen, 1997), 17-145, Springer, New York, 1999. MR 1678467 (2000b:37053)
  • 7. Khanin, K.; Kifer, Y., Thermodynamic formalism for random transformations and statistical mechanics, Sinai's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, 171, 107-140, Amer. Math. Soc., Providence, RI, 1996. MR 1359097 (96j:58136)
  • 8. Kifer, Y., Random dynamics and its applications, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math., 1998, Extra Vol. II, 809-818 (electronic). MR 1648128 (99k:58113)
  • 9. Kingman, J., Subadditive ergodic theory, Annals of Probability, 1 (1973) 883-904. MR 0356192 (50:8663)
  • 10. Liu, P. D., Dynamics of random transformations: smooth ergodic theory, Ergodic Theory Dynam. Systems, 21 (2001) 1279-1319. MR 1855833 (2002g:37024)
  • 11. Liu, P. D.; Zhao, Y., Large deviations in random perturbations of Axiom A basic sets, J. London Math. Soc, 68 (2003) 148-164. MR 1980249 (2006j:37059)
  • 12. Oseledec, V. I., A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Transactions of the Moscow Mathematical Society, 19, Amer. Math. Soc., Providence, RI, 1968. MR 0240280 (39:1629)
  • 13. Ruelle, D., Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2), 115 (1982) 243-290. MR 647807 (83j:58097)
  • 14. Slomczyński, W., Subadditive ergodic theorems in $ C(X)$, Ital. J. Pure Appl. Math. (1997) 17-28. MR 1666986 (99m:47008)
  • 15. Sturman, R.; Stark, J., Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity, 13 (2000) 113-143. MR 1734626 (2000m:37041)
  • 16. Wagner, D., Survey of measurable selection theorems, SIAM J. Control and Optimization, 15 (1977) 859-903. MR 0486391 (58:6137)

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Additional Information

Yongluo Cao
Affiliation: Department of Mathematics, Suzhou University, Suzhou 215006, Jiangsu, People’s Republic of China
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

DOI: https://doi.org/10.1090/S0002-9939-08-09347-7
Keywords: Skew-product, random maps, Lyapunov exponents
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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