Representations of toral automorphisms

• Published in: Topology and its applications Vol. 205; pp. 88 - 116
• Main Author: Schmidt, Klaus
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.06.2016
• Subjects: Homoclinic points; Hyperbolic and quasihyperbolic toral automorphisms; Symbolic representations; Symbolic representations; 37B10; 37A45; 11K16; Hyperbolic and quasihyperbolic toral automorphisms; Homoclinic points; 37C29
• ISSN: 0166-8641; 1879-3207
• DOI: 10.1016/j.topol.2016.01.022

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.