Even and odd primality of dynamical systems with invariant measure

Abstract

KEY WOADS: dynamical system, invariant measure, multiple mixing, joining theory.In this paper, by a dynamical system we mean an action ~ = {Tg : g E G} of a group G in a Lebesgue space (X, g), g(X) = 1 such that ~ preserves the measure g. We say that the measure u given on the Cartesian cube X" belongs to the class M(n -1, n), n > 2, if its projections on the (n -1)-dimensional faces of the cube X" are equal to p,,-l. We say that an action q~ has the property S(n -1, n) (or belongs to the class S(n -1, n)) if g" is the only measure of the class M(n -1, n) that is invariant under the action 9 | q~ | | q~ (n times). The properties S(n -1, n) were studied in [1][2][3][4][5][6]. The interest in these properties arises from some problems of joining theory (see [3]) and from the investigation of multiple mixing (if a commuting mixing system of the class S(n, n + 1) intermixes with multiplicity n -1, then it also intermixes with multiplicity n, (see [5]).) It is known (see [4,6]) that i) all (even) properties S(2p -1,2p), p > 1, are equivalent;ii) the property S(3, 4) implies the property S(2q, 2q + 1) for any q = 1,2, ....In this paper, we consider an example of noncommutative action ~ having the property S(2q, 2q + 1) but not having the property S(2p-1,2p). In this case we say that the operation 'IJ has the odd tensor primality. This terminology can be explained by -the following. The dynamical system 9 corresponds to the group ~ of unitary operators acting in the space L2(X, g). The odd tensor primality means that does not allow nontrivial Markov intertwinings with even tensor degree of the system ~.Let us define the action ~. First, note that the automorphisms of the group X = Z2 x Z2 x Z2 x -.. and the shifts on X preserve the Haar measure /~ on X. For a permutation a of the set of natural numbers N, we define an automorphism T~, of the group X by setting T~,({zi}) = {z~,(0}. We assign a shift S~(x) = z + a to the sequence a = {ai} E X ; here ' +' is the group operation in X. In the role of ~I', we consider the action generated by all T,, and S,,(z).

On simplicity concepts for ergodic actions

“…This means that we need only distinguish 3 categories of simplicity: 2-simplicity, 3-simplicity and simplicity. In general the three categories are mutually different-the corresponding examples are constructed in [Ry1] and [PRy]. However in the case of Z-actions, two of these categories coincide: a weakly mixing 3-fold simple transformation is simple [GlHR] (this holds also for Z n -actions).…”