Representations of toral automorphisms

TL;DR

This survey reviews algebraic symbolic representations of ergodic automorphisms on compact abelian groups, focusing on homoclinic points and extending concepts from hyperbolic to nonexpansive automorphisms, highlighting ongoing challenges.

Contribution

It summarizes the development of symbolic covers for automorphisms of compact abelian groups, including recent extensions beyond expansive cases.

Findings

Homoclinic points are central to constructing symbolic representations.

Extensions to nonexpansive automorphisms are possible but complex.

The survey discusses phenomena and open problems in these extensions.

Abstract

This survey gives an account of an algebraic construction of symbolic covers and representations of ergodic automorphisms of compact abelian groups, initiated by A.M. Vershik around 1992 for hyperbolic automorphisms of finite-dimensional tori. The key ingredient in this approach, which was subsequently extended to arbitrary expansive automorphisms of compact abelian groups, is the use of homoclinic points of the automorphism. Although existence and abundance of homoclinic points is intimately connected to expansiveness of the automorphism, it is nevertheless possible to extend certain aspects of this construction to nonexpansive irreducible automorphisms of compact abelian groups (like irreducible toral automorphisms whose dominant eigenvalue is a Salem number). The later sections of this survey discuss the phenomena and problems arising in this extension.