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Continuity properties of Lyapunov exponents for surface diffeomorphisms

Abstract

We study the entropy and Lyapunov exponents of invariant measures \(\mu \) for smooth surface diffeomorphisms f, as functions of \((f,\mu )\). The main result is an inequality relating the discontinuities of these functions. One consequence is that for a \(C^\infty \) surface diffeomorphism, on any set of ergodic measures with entropy bounded away from zero, continuity of the entropy implies continuity of the exponents. Another consequence is the upper semi-continuity of the Hausdorff dimension on the set of ergodic invariant measures with entropy bounded away from zero. We also obtain a new criterion for the existence of SRB measures with positive entropy.

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Correspondence to Sylvain Crovisier.

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Part of this work was done when O.S. was visiting Université Paris-Sud and IHÉS, and he would like to thank these institutions for their hospitality and excellent working conditions. O.S. also acknowledges partial support of ISF grant 1149/18.

J.B. was partially supported by the ISDEEC project ANR-16-CE40-0013.

S.C. was partially supported by the ERC project 692925 NUHGD

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Buzzi, J., Crovisier, S. & Sarig, O. Continuity properties of Lyapunov exponents for surface diffeomorphisms. Invent. math. (2022). https://doi.org/10.1007/s00222-022-01132-x

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  • DOI: https://doi.org/10.1007/s00222-022-01132-x

Mathematics Subject Classification

  • 37C40
  • 37D30
  • 37A35
  • 37D35