On dynamical systems with specification-like properties
April 9, 2014 [Wednesday]
1:00 pm 646 PGH
Abstract
I will present a survey of various properties similar to Bowen's
specification property. I will concentrate on their influence on the
simplex of invariant measures and the entropy function. I will describe a
new approach coming from my joint work with Katrin Gelfert. We have
introduced two topological conditions for a dynamical system: closeability
with respect to some set of periodic points and linkability of a set of
periodic points. Together they imply that the set of invariant measures of
a continuous map on a compact metric space is either a single periodic
orbit or the Poulsen simplex - the unique non-trivial Choquet simplex with
dense set of extreme points. These conditions generalize the periodic
specification property used previously to show that ergodic measures are
dense among all invariant measures. It turns out that all beta-shifts, all
S-gap shifts, and many other dynamical systems posses closeability and
linkability. These conditions also imply that every invariant measure has a
generic point and allow to prove results about generic properties of
invariant measures generalizing Sigmund's theorem. I will provide examples
which allow to distinguish between our approach and old and more recent
specification-like methods of Sigmund, Bowen, Climenhaga-Thompson,
Pfister-Sullivan. To this end I introduce a new class of shift spaces
generalizing S-gap shifts.
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Last modified: April 08 2016 - 20:30:35