J. Feldman

An amenable equivalence relation is generated by a single transformation

We prove that for any amenable non-singular countable equivalence relation R ⊂ X × X , there exists a non-singular transformation T of X such that, up to a null set: It follows that any two Cartan subalgebras of a hyperfinite factor are conjugate by an automorphism.

Orbit structure and countable sections for actions of continuous groups

Abstract It is shown that if a second countable locally compact group G acts nonsingularly on an analytic measure space ( S , μ ), then there is a Borel subset E ⊂ S such that EG is conull in S and each sG ∩ E is countable. It follows that the measure groupoid constructed from the equivalence relation s ∼ sg on E may be simply described in terms of the measure groupoid made from the action of some countable group. Some simplifications are made in Mackeys theory of measure groupoids. A natural notion of “approximate finiteness” ( AF ) is introduced for nonsingular actions of G , and results are developed parallel to those for countable groups; several classes of examples arising naturally are shown to be AF . Results on “skew product” group actions are obtained, generalizing the countable case, and partially answering a question of Mackey. We also show that a group-measure space factor obtained from a continuous group action is isomorphic (as a von Neumann algebra) to one obtained from a discrete group action.

NewK-automorphisms and a problem of Kakutani

AbstractA property is introduced, for 1-1 measure-preserving transformations of probability spaces, calledloose Bernoulliness (LB), which is invariant under taking factors, inducing, and tower-building. It amounts to replacing, in Ornstein’s definition ofvery weak Bernoulli, the Hamming distance on strings by a coarser metric. The main result is the construction of a transformationT0 which is ergodic and of entropy 0 butnot LB. On the other hand, any irrational rotationis LB. Consequently, the equivalence relation generated by inducing and tower-building (which I callKakutani equivalence, and the Russians callmonotone equivalence) has at least two distinct equivalence classes among the ergodic entropy zero transformations. A similar situation exists for ergodic positive-entropy transformations: on the one hand, any Bernoulli shift is LB, while on the other hand a non LBK-automorphism

r-entropy, equipartition, and Ornstein’s isomorphism theorem inR n

\hat T_0

Normal numbers from independent processes

can be made by skewingT0 over a Bernoulli base.

A generalization of a result of R. Lyons about measures on [0, 1)

We introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II 1 . Such pairs are characterized by a group Q which serves as a quotient for the pair, or by the ability to synthesize the larger relation from the smaller and an action (modulo inner automorphisms) of Q on it; in the case where Q is amenable, one can work with a genuine action. We classify ergodic subrelations of finite index, and arbitrary normal subrelations, of the unique amenable relation of type II 1 . We also give a number of rigidity results; for example, if an equivalence relation is generated by a free II 1 -action of a lattice in a higher rank simple connected non-compact Lie group with finite centre, the only normal ergodic subrelations are of finite index, and the only strongly normal, amenable subrelations are finite.

A ratio ergodic theorem for commuting, conservative, invertible transformations with quasi-invariant measure summed over symmetric hypercubes

A new approach is given to the entropy of a probability-preserving group action (in the context ofZ and ofRn), by defining an approximate “r-entropy”, 0<r<1, and lettingr → 0. If the usual entropy may be described as the growth rate of the number of essential names, then ther-entropy is the growth rate of the number of essential “groups of names” of width≦r, in an appropriate sense. The approach is especially useful for actions of continuous groups. We apply these techniques to state and prove a “second order” equipartition theorem forZm ×Rn and to give a “natural” proof of Ornstein’s isomorphism theorem for Bernoulli actions ofZm ×Rn, as well as a characterization of such actions which seems to be the appropriate generalization of “finitely determined”.

Countable sections for free actions of groups

Abstract Sequences of independent random variables and products of probability spaces are just two ways of looking at the same thing. The natural generalization of a sequence of independent random variables is a decomposable process. We introduce a corresponding generalization of a product of probability spaces, which will be called a factored probability space, and study the structure and classification of such systems and their relation to decomposable processes.