Abstract. Two properties of a dynamical system, rigidity and non-recurrence, are examined in detail. The ultimate aim is to characterize the sequences along which these properties do or do not occur for different classes of transformations. The main focus in this article is to characterize explicitly the structural properties of sequences which can be rigidity sequences or non-recurrent sequences for some weakly mixing dynamical system. For ergodic transformations generally and for weakly mixing transformations in particular there are both parallels and distinctions between the class of rigid sequences and the class of non-recurrent sequences. A variety of classes of sequences with various properties are considered showing the complicated and rich structure of rigid and non-recurrent sequences.
We call an ergodic measure-preserving action of a locally compact group G on a probability space simple if every ergodic joining of it to itself is either product measure or is supported on a graph, and a similar condition holds for multiple self-joinings This generalizes Rudolph's notion of minimal self-joinings and Veech's property S Main results The joinings of a simple action with an arbitrary ergodic action can be explicitly descnbed A weakly mixing group extension of an action with minimal self-joinings is simple The action of a closed, normal, co-compact subgroup in a weakly-mixing simple action is again simple Some corollaries Two simple actions with no common factors are disjoint The time-one map of a weakly mixing flow with minimal self-joinings is prime Distinct positive times in a Z-action with minimal self-joinings are disjoint 0 Introduction and definitions The notion of minimal self-joinings for Z-actions was introduced in [Ru2] as a source of counter-examples In this paper we generalize this notion to what we call simple group actions and develop some general theory for these actions This allows us to broaden the repertoire of actions displaying this sort of behaviour We deal with actions of fairly general groups because it is convenient for our purposes and not much more difficult, but the main interest lies in Z and R-actions Most of our results are new even within the setting of Z-actionsWe consider a standard Borel space (X, B), that is there exists a complete separable metric on X such that B = B(X) is the cr-algebra of Borel sets generated by the corresponding topology on X (By the remarks on p 138 of [Ma2] one can assume that the metric on X is actually compact) Suppose that X is equipped with a Borel probability measure /J, and that G is a locally compact group By a (left) action of G on X we mean a Borel map GxX->X denoted (g, x)>-*gx such that
(hg)x = h(gx)Vh,g€G,xeX, and ex = x Vx e X,
We define two families of relations between ergodic ℤn actions, both indexed equivariantly by non-singular n × n matrices. The first is to be Katok cross-sections of the same flow, indexed in a natural way by the matrices. The second is determined by the existence of an orbit preserving injection with an extra asymptotic linearity condition. We demonstrate that these two families are identical. In one dimension this is the classical theory of Kakutani equivalence.
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