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Article Dans Une Revue Inventiones Mathematicae Année : 2022

Continuity properties of Lyapunov exponents for surface diffeomorphisms

Jérôme Buzzi
Sylvain Crovisier
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  • PersonId : 995905
Omri Sarig
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  • PersonId : 847557

Résumé

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 diffeomorphisms, 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.

Dates et versions

hal-03158542 , version 1 (04-03-2021)

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Jérôme Buzzi, Sylvain Crovisier, Omri Sarig. Continuity properties of Lyapunov exponents for surface diffeomorphisms. Inventiones Mathematicae, 2022, 230, pp.767-849. ⟨hal-03158542⟩
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