# Invariant measures for Cantor dynamical systems

## TL;DR

This survey explores probability and infinite ergodic invariant measures for aperiodic homeomorphisms of Cantor sets, emphasizing cases with unique or finitely many ergodic measures, using combinatorial methods from symbolic dynamics.

## Contribution

It provides a comprehensive overview of invariant measures for Cantor dynamical systems, highlighting recent combinatorial approaches and classifications.

## Key findings

- Characterization of invariant measures for aperiodic homeomorphisms
- Analysis of systems with unique or finitely many ergodic measures
- Application of Bratteli diagrams and symbolic dynamics methods

## 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.