Symbolic dynamics for surface diffeomorphisms with positive topological entropy

Details Symbolic dynamics for surface diffeomorphisms with positive topological entropy Symbolic dynamics for surface diffeomorphisms with positive topological entropy

2011-05-09

arxiv.org/abs/1105.1650v3

Mathematics

Dynamical Systems

Added some references, corrected some typos

Scientific paper

Suppose f is a C^{1+\epsilon} surface diffeomorphism with positive topological entropy. For every positive \delta strictly smaller than the topological entropy of f we construct an invariant Borel set E such that (a) f|E has a countable Markov partition; and (b) E has full measure with respect to any ergodic invariant probability measure with entropy larger than \delta. This allows us to prove the following conjecture of A. Katok: if f is C^\infty with topological entropy h>0, and if P_n(f)=#{x:f^n(x)=x}, then limsup P_n(f)/exp(nh)>0.

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