ℤ d -actions on compact abelian groups

Abstract

According to Theorem 4.2, ℤd is of Markov type for every d ≥ 1, and ℤd-actions by automorphisms of compact groups enjoy the properties described in (4.10), Propositions 4.9-4.10, Remark 4.15, and Theorem 4.11. Just as compact, abelian groups like Tn = ℝn/ℤn have automorphisms with very intricate dynamical properties, there is an abundance of examples of interesting ℤd-actions by automorphisms of compact abelian groups. In this section we introduce a general formalism for the investigation of such actions which will also give us a systematic approach to constructing actions with specified properties.