Dynamical Systems of Algebraic Origin pp 285-300 | Cite as

# Rigidity

## Abstract

An ergodic ℤ^{ d }-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 ℤ^{ d } 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, ℤ^{ d }-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]).

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