Dynamics in dimension zero. A survey

TL;DR

This survey reviews techniques for analyzing zero-dimensional dynamical systems, including various representations, entropy, and minimal models, emphasizing marker techniques and introducing decisiveness of Bratteli-Vershik systems.

Contribution

It consolidates multiple methods and recent notions in zero-dimensional dynamics, providing new insights into representations, entropy, and minimal models, and introduces the concept of decisiveness.

Findings

Switching between representations is feasible and useful.

Marker techniques are effective in entropy and minimal model results.

Decisiveness of Bratteli-Vershik systems is characterized by a sufficient condition.

Abstract

The goal of this paper is to put together several techniques in handling dynamical systems on zero-dimensional spaces, such as array representation, inverse limit representation, or Bratteli-Vershik representation. We describe how one can switch from one representation to another. We also briefly review some more recent related notions: symbolic extensions, symbolic extensions with an embedding, and uniform generators. We devote a great deal of attention to marker techniques and we use them to prove two types of results: one concerning entropy and vertical data compression, and another, about the existence of isomorphic minimal models for aperiodic systems. We also introduce so-called decisiveness of Bratteli--Vershik systems and give for it a sufficient condition.