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A minimum principle for Lyapunov exponents and a higher-dimensional version of a theorem of Mañé

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Abstract

We consider compact invariant sets Λ forC 1 maps in arbitrary dimension. We prove that if Λ contains no critical points then there exists an invariant probability measure with a Lyapunov exponent λ which is theminimum of all Lyapunov exponents for all invariant measures supported on Λ. We apply this result to prove that Λ isuniformly expanding if every invariant probability measure supported on Λ is hyperbolic repelling. This generalizes a well known theorem of Mañé to the higher-dimensional setting.

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References

  1. J.F. Alves,Statistical Analysis of Nonuniformly Expanding Dynamical Systems, 2003, Preprint.

  2. J.F. Alves, V. Araujo, andB. Saussol,On the uniform hyperbolicity of some nonuniformly hyperbolic systems, Proc. Amer. Math. Soc.131(4) (2003), 1303–1309.

    Article  MATH  MathSciNet  Google Scholar 

  3. Y. Cao,Non-zero Lyapunov exponents and uniform hyperbolicity, Nonlinearity16 (2003), no. 4, 1473–1479.

    Article  MATH  MathSciNet  Google Scholar 

  4. R.A. Johnson, K.J. Palmer, andG.R. Sell,Ergodic properties of linear dynamical systems, SIAM J. Math. Anal.18 (1987), no. 1, 1–33.

    Article  MATH  MathSciNet  Google Scholar 

  5. S. Luzzatto,Stochastic behavour in non-uniformly expanding maps, Handbook of dynamical systems. Vol. 1B, 2006, pp. 265–326.

    Article  Google Scholar 

  6. R. Mañé,Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys.100 (1985) no. 4, 495–524.

    Article  MATH  MathSciNet  Google Scholar 

  7. T. Nowicki andD. Sands,Non-uniform, hyperbolicity and universal bounds for S-unimodal maps, Invent. Math.132 (1998), no. 3, 633–680.

    Article  MATH  MathSciNet  Google Scholar 

  8. V.I. Oseledec,A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Transactions of the Moscow Mathematical Society19 (1968).

  9. D. Ruelle,Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2)115 (1982), no. 2, 243–290.

    Article  MathSciNet  Google Scholar 

  10. W. Slomczyński,Continuous subadditive processes and formulae for Lyapunov characteristic exponents, Ann. Polon. Math.61 (1995), no. 2, 101–134.

    MATH  MathSciNet  Google Scholar 

  11. W. Slomczyśki,Subadditive ergodic theorems in C(X), Ital. J. Pure Appl. Math. (1997), no. 1, 17–28 (1998).

    Google Scholar 

  12. R. Sturman andJ. Stark,Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity13 (2000), no. 1, 113–143.

    Article  MATH  MathSciNet  Google Scholar 

  13. P. Walters,An Introduction to ergodic theory, Springer-Verlag, New York, Berlin, Heidelberg, 1982.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics Dept, Suzhou University, 215006, Suzhou Jiangsu, P. R. China

    Yongluo Cao

  2. Mathematics Dept, Imperial College, 180 Queen's Gate, SW7 2AZ, London, UK

    Stefano Luzzatto

  3. Mathematics Dept, Universidade Federal Fluminense, Niteroi, RJ, Brazil

    Isabel Rios

Authors
  1. Yongluo Cao
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  2. Stefano Luzzatto
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  3. Isabel Rios
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Corresponding author

Correspondence to Yongluo Cao.

Additional information

The authors acknowledge the hospitality of Imperial College London where this work was carried out. They also thank Jaroslav Stark for several useful comments on a previous version of the paper and for pointing out several relevant references.

partially supported by NSF (10071055) and SFMSBRP of China and The Royal Society

partially supported by CAPES and FAPERJ (Brazil)

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Cao, Y., Luzzatto, S. & Rios, I. A minimum principle for Lyapunov exponents and a higher-dimensional version of a theorem of Mañé. Qual. Th. Dyn. Syst 5, 261–273 (2004). https://doi.org/10.1007/BF02972681

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