Self-joinings of commutative actions with invariant measure

Abstract

In the investigation of a dynamical system with invariant measure, its interaction with other systems and its arrangement in the semigroup of Markovian operators are of interest. Problems concerning the centralizer of an action and on the structure of its factors are special cases of this general problem. Joinings are a useful tool in the study. Transformations with minimal self-joinings were introduced by Rudolf in ergodic theory as an instrument to construct actions with diverse unusual properties. In particular, transformations with minimal self-joinings have no factors, and their centralizers are trivial. This stimulated the investigation of the more general class of so-called simple systems (see, e.g., [1]–[3]). Let T : X → X be a measure-preserving invertible transformation of a measure space (X,B, μ), μ(X) = 1. A measure ν on X(1) × · · · ×X(n) is referred to as a self-joining of order n if it is T(1) × · · · × T(n)-invariant and its projections to X(i) are equal to μ. In the definition of self-joining for a group action {Tg : g ∈ G} it is assumed that the measure ν is Tg × · · · × Tg-invariant for any g. The image of the measure μ under the mapping φS : X → X ×X, where φS(x) = (x, S(x)), is denoted by ΔS (a shift of the diagonal measure on X ×X). Note that the measure ΔS is defined by the ruleΔS(A×B) = μ(SA ∩B). If an automorphism S commutes with the action {Tg : g ∈ G}, then ΔS is a self-joining of order 2. An action {Tg} is said to be 2-simple if any ergodic self-joining of order 2 either is of the formΔS or coincides with the measure μ× μ. For any 2-simple system, the extreme points of the semigroup of all Markovian operators commuting with the system are the automorphisms commuting with the system and the operator Θ defining the orthogonal projection to the space of constants in L2(μ). An action {Tg} is said to be n-simple if any ergodic self-joining ν of order n of the action either coincides with the measure μn or has a projection (of the measure ν) to some two-dimensional face of the cubeX1 × · · · ×Xn which is of the formΔS . The investigation of self-joinings of higher order is closely related to Rokhlin’s problem on multiple mixing. In the class of automorphisms with minimal Markovian centralizer, the existence of a nontrivial self-joining of order 3 implies the solution of this famous problem (see [3]) and, for a mixing automorphism of positive local rank, gives both the solution of the Rokhlin problem and the solution of the weakened Banach problem on the absolutely continuous spectrum of finite multiplicity (see the references in [4]). For these reasons (and some others), the problem of coincidence of the 2-simplicity of a commutative action with the property of higher-order simplicity is central in the theory of self-joinings. The action of a direct sum of finite groups was studied in spectral theory by Stepin [5]. The growth of interest to dynamical systems with nonclassical time is a tendency of recent years (see the survey [6]). Group actions are fruitfully applied to the investigation of individual automorphisms entering these actions ([2], [7], [8]).