On density of ergodic measures and generic points
Abstract
We provide conditions which guarantee that ergodic measures are dense in the simplex of invariant probability measures of a dynamical system given by a continuous map acting on a Polish space. Using them we study generic properties of invariant measures and prove that every invariant measure has a generic point. In the compact case, density of ergodic measures means that the simplex of invariant measures is either a singleton of a measure concentrated on a single periodic orbit or the Poulsen simplex. Our properties focus on the set of periodic points and we introduce two concepts: close\ability with respect to a set of periodic points and linkability of a set of periodic points. Examples are provided to show that these are independent properties. They hold, for example, for systems having the periodic specification property. But they hold also for a much wider class of systems which contains, for example, irreducible Markov chains over a countable alphabet, all $\beta$shifts, all $S$gap shifts, ${C}^1$generic diffeomorphisms of a compact manifold $M$, and certain geodesic flows of a complete connected negatively curved manifold.
 Publication:

arXiv eprints
 Pub Date:
 April 2014
 DOI:
 10.48550/arXiv.1404.0456
 arXiv:
 arXiv:1404.0456
 Bibcode:
 2014arXiv1404.0456G
 Keywords:

 Mathematics  Dynamical Systems;
 37B05;
 37B10;
 37A99;
 37D25;
 37C20
 EPrint:
 32 pages, 6 figures. This version replaces an earlier preprint entitled "The (Poulsen) simplex of invariant measures"