Symbolic dynamics for surface diffeomorphisms with positive entropy

Abstract

Letffbe aC1+εC^{1+\varepsilon }diffeomorphism on a compact smooth surface with positive topological entropyhh. For every0>δ>h0>\delta >h, we construct an invariant Borel setEEand a countable Markov partition for the restriction offftoEEin such a way thatEEhas full measure with respect to every ergodic invariant probability measure with entropy greater thanδ\delta. The following results follow:ffhas at most countably many ergodic measures of maximal entropy (a conjecture of J. Buzzi), and in the case whenffisC∞C^\infty,lim supn→∞e−nh#{x:fn(x)=x}>0\limsup \limits _{n\to \infty }e^{-n h}\#\{x:f^n(x)=x\}>0(a conjecture of A. Katok).