On full groups of measure-preserving and ergodic transformations with uncountable cofinalities

The group of all measure-preserving permutations of the unitinterval and the full group of an ergodic transformation ofthe unit interval are shown to have uncountable cofinality andthe Bergman property. Here, a group G is said to have the Bergmanproperty if, for any generating subset E of G, some boundedpower of EE^–1{1} already covers G. This property arosein a recent interesting paper of Bergman, where it was derivedfor the infinite symmetric groups. We give a general sufficientcriterion for groups G to have the Bergman property. We showthat the criterion applies to a range of other groups, includingsufficiently transitive groups of measure-preserving, non-singular,or ergodic transformations of the reals; it also applies tolarge groups of homeomorphisms of the rationals, the irrationals,or the Cantor set.     

Commuting measure-preserving transformations

Let φ₁, ... ,φ _d be commuting measure-preserving transformations, . The Kakutani-Rokhlin tower theorem is proved in a refined form for non-periodic groups Φ, and the Shannon-McMillan theorem is extended to ergodic groups. These results are used to extend recent isomorphism results to groups of transformations.     

On circle-valued cocycles of an ergodic measure-preserving transformation

Analytic necessary and sufficient conditions are given for a circle-valued functionf to generate a cocycle which is a multiple of a coboundary. These conditions are then used to derive some other new criteria for cocycles to be coboundaries. This research was supported in part by an NSF grant DMS8600753.     

Spectra of ergodic transformations

We study the group properties of the spectrum of a strongly continuous unitary representation of a locally compact Abelian group G implementing an ergodic group of ¹-automorphisms of a von Neumann algebra $\text{R}$. It is shown that in many cases the spectrum equals the dual group of G; e.g. if G is the integers and $\text{R}$ not finite dimensional and Abelian, then the spectrum is the circle group.     

Rigid factors of ergodic transformations

We prove a theorem concerning cartesian products of ergodic not necessarily measuring preserving transformations, using the notion of rigid factors for such transformations.     

Generators of an Abelian group of invertible measure-preserving transformations

A multi-dimensional counterpart of the asymptotic rate is introduced as a new conjugacy invariant for finitely generated Abelian groups of automorphisms of a fixed probability space. The generalization of the Kolmogorov generator theorem as well as some related results are established. It is not assumed that the underlying probability space is a Lebesgue one.     

Strictly ergodic models for non-invertible transformations

We generalize a result of R. Jewett [J]: IfT is an ergodic measure preserving transformation on (X, Ω,λ),T not necessarily invertible, there exists a strictly ergodicS acting on (Y, Θ,ν), whereY is compact, such that (X, Ω,λ, T) is measure theoretically isomorphic to (Y, Θ,ν, S).     

Undecidability of the elementary theory of groups of measure-preserving transformations

The undecidability of the elementary theory of the automorphism group for a Lebesgue space is proved. It is shown that arithmetic can be interpreted in this theory. The technique of proof can be carried over to certain other groups. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 414–420, March, 1998. The author wishes to thank the participants of the seminar on dynamical systems headed by D. V. Anosov, R. I. Grigorchuk, and A. M. Stepin as well as the participants of the Kolmogorov seminar of the Mechanics and Mathematics Department of Moscow State University for discussion of this work. The author is also indebted to N. K. Vereshchagin and A. M. Stepin for support and valuable remarks as well as to V. V. Ryzhikov for setting the problem and for his assistance.     

Abelian cocycles for nonsingular ergodic transformations and the genericity of type III1 transformations

The authors prove that in the space of nonsingular transformations of a Lebesgue probability space the type III₁ ergodic transformations form a denseG set with respect to the coarse topology. They also prove that for any locally compact second countable abelian groupH, and any ergodic type III transformationT, it is generic in the space ofH-valued cocycles for the integer action given byT that the skew product ofT with the cocycle is orbit equivalent toT. Similar results are given for ergodic measure-preserving transformations as well.Research supported in part by: Nat. Sci. and Eng. Res. Council #A7163 and # U0080 F.C.A.C. Quebec, NSF Grants # MCS-8102399 and # DMS-8418431.     

Ergodicity and weak-mixing of homogeneous extensions of measure-preserving transformations with applications to Markov shifts

Given a homogeneous extensionS of a measure-preserving transformationT, we provide necessary and sufficient conditions for the ergodicity and weak-mixing ofS in terms of functional equations. We then apply our findings to the case whenT is a Markov shift and the associated skewing function ofS depends on a finite number of coordinates. In this case, we obtain a simplification to the appropriate functional equations.