Tag Archives: space of invariant measures

Equidistribution for random rotations

Two very different types of dynamical behaviour are illustrated by a pair of very well-known examples on the circle: the doubling map and an irrational rotation. On the unit circle in , the doubling map is given by , while … Continue reading

Posted in ergodic theory, examples, random dynamics, statistical laws | Tagged | Leave a comment

A useful example for the space of ergodic measures

Last time I discussed the following three properties that may or may not be satisfied for a map on a compact metric space : (C) The set of ergodic measures is path-connected. (D) is dense in the set of all … Continue reading

Posted in ergodic theory, examples, topological dynamics | Tagged , , , | 1 Comment

Some questions on ergodic measures

This post will be a brief set of notes recording some thoughts from my talk at the Penn State dynamics seminar today, since I did not produce slides but rather spoke more informally about some results I’ve recently learned (which … Continue reading

Posted in ergodic theory, topological dynamics | Tagged , , | 1 Comment

Specification

In the last post we saw that if is the space of invariant probability measures for the full shift, then the collection of ergodic measures (which are the extreme points of the simplex ) has two remarkable properties: is dense … Continue reading

Posted in ergodic theory, topological dynamics | Tagged , , | 2 Comments

Notions of irreducibility

Let be a topological dynamical system. (Generally this means, for me at least, a continuous self-map of a compact metric space. However, sometimes one may be interested in examples that are not compact or that are only piecewise continuous.) We … Continue reading

Posted in ergodic theory, topological dynamics | Tagged , | 2 Comments