Robustness of the measure of maximal entropy for smooth surface diffeomorphisms

Speaker. Omri Sarig
Date. 19.12.23 at 14:30

Abstract. Let f be a topologically transitive C-infinity surface diffeomorphism with positive topological entropy. Let m be the (unique) measure of maximal entropy. We show that for any invariant measure m’ whose entropy is epsilon
away from the maximal entropy,

(1) |m(h)-m'(h)|=O(\sqrt{\epsilon}) for all Holder functions h with unit Holder norm
(2) the Lyapunov exponents of m’ are O(\sqrt{\epsilon}) away from those of m
(3) the Oseledets splitting of m’ is O(\sqrt{\epsilon}) away from that of m’ (in a sense that will be made precise in the talk).

The case of Anosov diffeomorphisms is due to Kadirov; the novelty of our work is that we do not need the Anosov assumption. (Joint with Jerome Buzzi and Sylvain Crovisier)