Symbolic dynamics for surface diffeomorphisms with positive entropy

Source abstract

Let f f be a C 1 + ε C^{1+\varepsilon } diffeomorphism on a compact smooth surface with positive topological entropy h h . For every 0 > δ > h 0>\delta >h , we construct an invariant Borel set E E and a countable Markov partition for the restriction of f f to E E in such a way that E E has full measure with respect to every ergodic invariant probability measure with entropy greater than δ \delta . The following results follow: f f has at most countably many ergodic measures of maximal entropy (a conjecture of J. Buzzi), and in the case when f f is C ∞ C^\infty , lim sup n → ∞ e − n h # { x : f n ( x ) = x } > 0 \limsup \limits _{n\to \infty }e^{-n h}\#\{x:f^n(x)=x\}>0 (a conjecture of A. Katok).