ALGEBRAIC Zd-ACTIONS ON ZERO-DIMENSIONAL COMPACT ABELIAN GROUPS

In 1967 Furstenberg proved that every infinite closed subset of T = R/Z simultaneously invariant under multiplication by 2 and by 3 is equal to T (cf. [8]), which motivated the still unresolved question whether this scarcity of invariant sets is paralleled by a corresponding scarcity of invariant probability measures: is Lebesgue measure the only nonatomic probability measure on T which is invariant under multiplication by 2 and by 3? Furstenberg’s question remained dormant until 1988, when Lyons [18] proved that any probability measure on T which has completely positive entropy under either of these maps is equal to Lebesgue measure. In 1990 Rudolph weakened Lyons’ hypotheses and proved the same result for any probability measure which is invariant and ergodic under the N2-action generated by multiplication by 2 and by 3 and has positive entropy under either of these maps. In 1996 Katok and Spatzier [10] introduced a remarkable extension of the scope of Furstenberg’s question to certain Zd-actions by automorphisms of compact abelian groups with d > 1.1 They proved that any probability measure μ on a finite-dimensional torus X = Tn which is invariant and mixing under a topologically mixing, irreducible (Definition 4.2) and expansive2 algebraic Zd-action α, and which has positive entropy under some element of the action, is a translate of Lebesgue measure on an α-invariant subtorus of X (the hypotheses in [10] are actually much weaker, but somewhat technical). The definitive version of this result is due to Einsiedler and Lindenstrauss [6] and implies that, for any probability measure μ on a finite-dimensional torus or solenoid X which is invariant and weakly mixing under a topologically mixing algebraic Zd-action α, there exists a closed α-invariant subgroup Y ⊂X such that μ = μ∗λY and the action induced by each αn, n∈Zd , on X/Y has zero entropy with respect to the measure μ̄ = μπ−1 (cf. Theorem 5.3). Here π : X −→ X/Y is the quotient map. Instead of pursuing further the many fascinating extensions of these measure rigidity results due to Katok and others let me turn to isomorphism rigidity of algebraic Zd-actions. Suppose that α and β are topologically mixing algebraic Zd-actions on finite-dimensional tori or solenoids X and Y , respectively. By following a suggestion of Thouvenot and applying the results in [10] or [6] to