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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.F. Alves,Statistical Analysis of Nonuniformly Expanding Dynamical Systems, 2003, Preprint.
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.
Y. Cao,Non-zero Lyapunov exponents and uniform hyperbolicity, Nonlinearity16 (2003), no. 4, 1473–1479.
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.
S. Luzzatto,Stochastic behavour in non-uniformly expanding maps, Handbook of dynamical systems. Vol. 1B, 2006, pp. 265–326.
R. Mañé,Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys.100 (1985) no. 4, 495–524.
T. Nowicki andD. Sands,Non-uniform, hyperbolicity and universal bounds for S-unimodal maps, Invent. Math.132 (1998), no. 3, 633–680.
V.I. Oseledec,A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Transactions of the Moscow Mathematical Society19 (1968).
D. Ruelle,Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2)115 (1982), no. 2, 243–290.
W. Slomczyński,Continuous subadditive processes and formulae for Lyapunov characteristic exponents, Ann. Polon. Math.61 (1995), no. 2, 101–134.
W. Slomczyśki,Subadditive ergodic theorems in C(X), Ital. J. Pure Appl. Math. (1997), no. 1, 17–28 (1998).
R. Sturman andJ. Stark,Semi-uniform ergodic theorems and applications to forced systems, Nonlinearity13 (2000), no. 1, 113–143.
P. Walters,An Introduction to ergodic theory, Springer-Verlag, New York, Berlin, Heidelberg, 1982.
Author information
Authors and Affiliations
Corresponding author
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)
Rights and permissions
About this article
Cite this article
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
Issue Date:
DOI: https://doi.org/10.1007/BF02972681