Invertible and noninvertible symbolic dynamics and their C*-algebras

TL;DR

This paper reviews recent progress in understanding how symbolic dynamical systems, both invertible and noninvertible, relate to C*-algebras, highlighting how system properties can be encoded and recovered through algebraic structures.

Contribution

It provides a comprehensive survey of how conjugacies and orbit equivalences of symbolic systems are represented within C*-algebras, including examples and open problems.

Findings

Encoding of conjugacies and orbit equivalences into C*-algebras

Recovery of dynamical systems from *-isomorphisms of C*-algebras

Illustrative examples and identification of open problems

Abstract

This paper surveys the recent advances in the interactions between symbolic dynamics and C*-algebras. We explain how conjugacies and orbit equivalences of both two-sided (invertible) and one-sided (noninvertible) symbolic systems may be encoded into C*-algebras, and how the dynamical systems can be recovered from structure-preserving *-isomorphisms of C*-algebras. We have included many illustrative examples as well as open problems.