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4:00 pm Friday, November 2, 2012 Dynamics LTL seminar: Sarig's construction of countable Markov partitions using Pesin theoryby Vaughn Climenhaga (University of Houston) in HB227- One of the fundamental constructions in hyperbolic dynamics is the semi-conjugacy between uniformly hyperbolic (Axiom A) systems and topological Markov chains on finite alphabets, an idea which goes back to work of J. Hadamard and E. Artin on geodesic flows, and was carried out in its greatest generality by Ya. Sinai and R. Bowen in the 1970s. Recently O. Sarig has given a version of this construction for the much broader class of surface diffeomorphisms with positive topological entropy, which uses Pesin theory (non-uniform hyperbolicity) to construct a countable Markov partition that captures the dynamics of every measure with sufficiently large entropy. In particular this allows Sarig to resolve conjectures due to A. Katok and to J. Buzzi. In this talk I will review the classical construction and explain the key innovations introduced by Sarig to deal with the non-uniformities present beyond the Axiom A setting.
Submitted by dani@rice.edu |