Rigidity

Abstract

An ergodic ℤd-action α by automorphisms of a compact, abelian group X usually has many distinct non-atomic, invariant, ergodic probability measures. For example, if d ≥ 1, and if α is the shift-action (2.1) of ℤd on \( X = {G^{{{\mathbb{Z}^2}}}} \mu = {v^{{{\mathbb{Z}^d}}}} \mu ' = v{'^{{{\mathbb{Z}^d}}}} \), where G is a nontrivial, compact, abelian group, then there exist uncountably many distinct, α-invariant, mixing probability measures on X: for every probability measure v on G, \( \mu = {v^{{{\mathbb{Z}^d}}}} \) is an α-invariant probability measure which is mixing of every order, and different measures v, v’ lead to inequivalent probability measures \( \mu = {v^{{{\mathbb{Z}^d}}}} \) and \( \mu ' = v{'^{{{\mathbb{Z}^d}}}} \). There exist, however, ℤd-actions by automorphisms of compact, abelian groups for which Haar measure is the only non-atomic, invariant, mixing probability measure, or the only invariant and ergodic probability measure on X with respect to which any α n has positive entropy (cf. Example 29.6 (1) and [89]).