Feldman-Katok pseudometric F-bar and the Gorodetski, Ilyashenko, Kleptsyn, and Nalsky (GIKN) construction of nonhyperbolic ergodic measures.
Expositor: Dominik Kwietniak
Instituição: Jagiellonian University e UFRJ
Data e Horário: 07/02/17 | 17:00h
RESUMO: The GIKN construction was introduced by Gorodetski, Ilyashenko, Kleptsyn, and Nalsky in 2005. It gives a nonhyperbolic ergodic measure which is a weak* limit of a special sequence of measures supported on periodic orbits. This method was later adapted by numerous authors and provided examples of nonhyperbolic invariant measures in various settings. We prove that the result of the GIKN construction is always a loosely Kronecker measure in the sense of Ornstein, Rudolph, and Weiss (equivalently, standard measure in the sense of Katok, another name is loosely Bernoulli measure with zero entropy). For a proof we introduce and study the Feldman-Katok pseudometric F-bar. The pseudodistance F-bar is a topological counterpart of the f-bar metric for finite-state stationary stochastic processes introduced by Feldman and, independently, by Katok, and later developed by Ornstein, Rudolph, and Weiss. We show that every measure given by the GIKN construction is the F-bar-limit of a sequence of periodic measures. On the other hand we prove that a measure which is the F-bar-limit of a sequence of ergodic measures is ergodic and its entropy is smaller or equal than the lower limit of entropies of measures in the sequence. Furthermore we demonstrate that F-bar-Cauchy sequence of periodic measures tends in the weak* topology to a loosely Kronecker measure.