Dynamical Systems Seminar

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.