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Tuesday 10 May 2016Dynamics seminar: On entropy-density of ergodic measures for B-free shifts and their relatives

Dominik Kwietniak - Jagiellonian University, Krakow

Laver LT6 13:30-14:30

We study properties of invariant measures of hereditary shift spaces, with an emphasis on B-free shifts (symbolic dynamical systems associated with B-free numbers). A shift space over {0,1} is hereditary if it is closed with respect to coordinatewise multiplication by 0-1 sequences. Given a subset of positive integers B, we say that an integer is B-free if it is not divisible by any element of B. For example square-free integers are {p^2:p-prime}-free. Sarnak proposed to study square-free numbers through dynamics of the shift space generated by the characteristic sequence of square-free numbers. Lemanczyk and his collaborators extended this idea to B-free shifts.

We prove that the set of ergodic measures is arcwise connected for every hereditary shift space and ergodic measures are entropy-dense if the shift can be approximated in the dbar sense by sofic shifts. Here dbar pseudodistance measures the upper asymptotic density of the set of positions where two sequences in a shift space differ. Our theory applies to many B-free shifts and their relatives which I am going to describe during the talk.

(This is a joint work with Jakub Konieczny and Michal Kupsa).

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