Algebra, Arithmetic and Multi-parameter Ergodic Theory

While classical ergodic theory deals largely with single ergodic transformations or flows (i.e. with actions of N,Z,R+ or R on measure spaces), many of the lattice models in statistical mechanics (such as dimer models) have multi-dimensional symmetry groups: they carry actions of Z or R with d > 1. However, the transition from Zor R-actions to multi-parameter ergodic theory presents considerable difficulties, even if one restricts attention to actions of Z with d ≥ 1 (as we shall do throughout this article). To illustrate this point, compare the classical theory of topological Markov chains (cf. e.g. [31]) with the complexities and undecidability problems arising in the study of cellular automata and more general multi-dimensional shifts of finite type (cf. [3], [49] or [24]). Even if undecidability is not an issue, multi-dimensional shift of finite type exhibit a markedly more complicated behaviour than their classical relatives (cf. e.g. [10, 11, 36, 41]). Another feature of the transition from d = 1 to d > 1 is that smooth Z-actions with d > 1 on compact manifolds have zero entropy, since individual elements of Z act with finite entropy. The powerful ideas and tools of smooth ergodic theory are thus of limited use for Z-actions. Furthermore, smooth Z-actions are not exactly abundant: all known examples arise from ‘algebraic’ constructions (commuting group translations, commuting automorphisms of finite-dimensional tori or solenoids, or actions of Cartan subgroups of semisimple Lie groups on homogeneous spaces). Again one should compare this with the richness of examples in classical smooth ergodic theory which contributes so much to the appeal of the subject. Making a virtue out of necessity, let us briefly turn to commuting automorphisms of finite-dimensional tori. Toral automorphisms are among the longest and most intensively studied measure-preserving transformations (their investigation contributed much to the formulation and understanding of fundamental dynamical concepts like hyperbolicity and geometrical notions of entropy), and it came as a considerable surprise when Hillel Furstenberg [19] proved in 1967 that unexpected things may happen if one studies not one, but two commuting toral maps: he showed that the only closed infinite subset of the circle T = R/Z which is simultaneously invariant under multiplication by 2 and by 3 is the circle itself (this is a statement about commuting surjective homomorphisms of T, but it has an immediate extension to commuting automorphisms of the 6-adic solenoid). In contrast, each of the two maps consisting of multiplication by 2 and by 3, respectively, is very easily seen to have many infinite closed invariant subsets. In connection with this result Furstenberg asked the famous — and still unanswered — question whether Lebesgue measure is the only nonatomic probability measure on T which is simultaneously invariant under multiplication by 2 and by 3. A partial answer to Furstenberg’s question was given by D. Rudolph in [38], where he showed that Lebesgue measure is the only nonatomic probability measure on T which is