Disjointness, divisibility, and quasi-simplicity of measure-preserving actions

Abstract

For weakly mixing flows, quasi-simplicity of order 2 implies quasi-simplicity of all orders. A uniformly divisible automorphism and a 2-quasi-simple automorphism are disjoint.Key words: weakly mixing flow, divisible ergodic system, quasi-simplicity, disjointness, pairwise independent joinings.The structure of self-joinings of a dynamical system gives information about the factors and the centralizer of its tensor products; it is also useful for the construction of counterexamples and for the study of Rokhlin's multiple mixing problem. A system that does not admit nontrivial pairwise independent self-joinings has the multiple mixing property. This leads us to the following question: what can one say about higher-order self-joinings if the structure of small-order self-joinings is known? There have been some progress for systems that commute only with polymorphisms of sufficiently simple structure (see [2]), for example, with convex sums of automorphisms and the orthoprojection operator. In this note, we consider a new class of actions, which commute with polymorphisms of more complicated structure. For these actions, which we call quasi-simple, we establish the above-mentioned property of absence of nontrivial higher-order self-joinings. This generalizes several results in [2], [3], and [7]. We also establish that a quasi-simple action is disjoint (in the sense of Furstenberg) from a Gaussian automorphism. (Note that their spectra are not necessarily disjoint.) This generalizes the results in [4] and [9].1. Let T and T be automorphisms of the spaces (X, µ) and (X , µ), respectively,. A measure ν on X × X whose marginals are equal to µ (i.e., a polymorphism in the terminology of [1]) is called a joining of the systems (T, X, µ) and (T , X, µ) if it is invariant with respect to T × T . A joining of flows T t and T t is defined in a similar way, but here the invariance with respect to T t × T t for all t ∈ R is assumed.We say that automorphisms T and T are disjoint if they admit only the (trivial) joining µ × µ. This means that there exists a unique Markov operator intertwining the unitary operators corresponding to T and T . This intertwining operator is the orthoprojection Θ onto the space of constant functions on (X, µ).A system (T, X, µ) is said to be 2-quasi-simple if for each ergodic self-joining ν = µ × µ the system (T ×T, X ×X, ν) is an isometric extension of either of its coordinate factors. If the coordinate factors coincide for each ergodic self-joining ν = µ × µ, then the system is said to be 2-simple.An "isometric extension" of a transformation is an extension with relatively discrete spectrum [10]. For example, a system is 2-quasi-simple if each ergodic self-joining ν = µ × µ of this system lies on the graph of a finite-valued map. (In this case, the conditional measures ν x are finite sums of point measures.) The latter property holds for flows possessing Ratner's property [6].Below we use only the following property of isometric extensions: if a system is generated by a weakly mixing factor F 1 and er...

Multiple mixing for a class of conservative surface flows

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Multiple mixing and disjointness for time changes of bounded-type Heisenberg nilflows

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Multiple mixing and parabolic divergence in smooth area-preserving flows on higher genus surfaces

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