Symbolic dynamics for surface diffeomorphisms with positive entropy

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).

Periodic expansiveness of smooth surface diffeomorphisms and applications

“…First recall we have by Katok's theorem [36] and its noninvertible version [22,49]: Therefore any C ∞ surface diffeomorphism or interval map f is asymptotically Per δ -expansive and satisfies g Per δ ≥ h top (f ) for any 0 < δ < h top (f ). Stronger lower bounds follow from the Markov representations built in [47], [18] and from Gurevic's theory: The main statements of the Introduction follow then directly from Corollary 2.1. A symbolic extension is a topological extension by a subshift over a finite alphabet (a symbolic extension does not need to be Markovian and the extension does not have to be finite-to-one).…”

“…The above results must be compared with the recent work of O. Sarig about the coding of C 1+α surface diffeomorphisms. In [47] he built for any δ > 0 a finite to one topological Markov shift extension of any surface diffeomorphism on a set of full measure for any hyperbolic ergodic measure with Lyapunov exponents δ-away from zero. Then the finite subgraphs of this Markov shift correspond to the hyperbolic sets.…”

“…[47][18](See Appendix A) Let f be a C r surface diffeomorpshim (or interval map) with r > 1 admitting a measure of maximal entropy. Then for any 0 < δ < h top there is a positive integer p such that lim inf n→+∞, p|n e −nhtop(f ) Per δ n > 0.…”

On a measure with maximal entropy for a suspension flow over a countable alphabet Markov shift

“…Suspension constructions (flows and cascades) over countable alphabet Markov shifts turn out to be a powerful tool in the study of some classes of smooth dynamical systems on manifolds [1,2,6,10].…”

A lower estimate of the entropy of an automorphism and maximum entropy conditions for and invariant measure of a suspension flow over a Markov shift

“…The topological (symbolic) Markov shifts, both with finite and countable alphabet, as well as suspen sions with discrete and continuous time over them, that is, suspension automorphisms (cascades) and sus pension flows, have long been important tools for studying many questions of the theory of smooth dynamical systems (see, e.g., [1][2][3][4][5][6]). We are interested in a lower estimate for the entropy of a flow of the type specified above corresponding to an invariant mea sure.…”