Invariant measures for Cantor dynamical systems

Abstract

This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure or finitely many such measures (finitely ergodic homeomorphisms). Since every Cantor dynamical system $(X,T)$ can be realized as a Vershik map acting on the path space of a Bratteli diagram, we use combinatorial methods developed in symbolic dynamics and Bratteli diagrams during the last decade to study the simplex of invariant measures.

37 pages, 2 figures. arXiv admin note: text overlap with arXiv:1709.00055, arXiv:1503.03360

References in corpus (3)

• Subdiagrams and invariant measures on Bratteli diagrams
• On the number of ergodic measures for minimal shifts with eventually constant complexity growth
• Bratteli--Vershik models and graph covering models