Invertible and noninvertible symbolic dynamics and their C*-algebras

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.