On the integrability of the ergodic hilbert transform for a class of functions with equal absolute values

It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable functionf such that the ergodic Hilbert transform of any function equal in absolute values tof is non-integrable.     

A counter-example to Dye's Theorem for all non-separable measure algebras

Summary In 1959, H. Dye showed that any two ergodic, measure-preserving automorphisms of a Lebesgue measure algebra were weakly equivalent. In this paper, we study weak equivalence, for ergodic measure-preserving automorphisms on non-separable measure algebras. It is shown that, in general, Dye's Theorem does not hold, and in particular, it holds only on separable, i.e. Lebesgue, measure algebras.     

Cutting and stacking,interval exchanges and geometric models

Every aperiodic measure-preserving transformation can be obtained by a cutting and stacking construction. It follows that all such transformations are infinite interval exchanges. This in turn is used to represent any ergodic measure-preserving flow as aC ^∞-flow on an open 2-manifold. Several additional applications of the basic theorems are also given. Partial support for this work was given by the National Science Foundation under grant number MCS81-07092.     

On the simplicity of the full group of ergodic transformations

LetE denote an invertible, non-singular, ergodic transformation of (0, 1). Then the full group ofE is perfect. IfE preserves the Lebesgue measure, then the full group is simple. IfE preserves no measure equivalent to Lebesgue, then the full group is simple. IfE preserves an infinite measure, then there exists a unique normal subgroup. IfT is any invertible transformation preserving the Lebesgue measure, then the full group is simple if and only ifT is ergodic on its support.     

Skew Products and Ergodic Theorems for Group Actions

New ergodic theorems for the action of a free semigroup on a probabilistic space by measure-preserving maps are obtained. The method applied consists of associating with the original semigroup action a skew product over the shift on the space of infinite one-sided sequences of generators of the semigroup and then integrating the BirkhoffKhinchin ergodic theorems along the base of the skew product. Bibliography: 17 titles.     

Some extremely amenable groupsQuelques groupes extrêmement moyennables

A topological group G is extremely amenable if every continuous action of G on a compact space has a fixed point. Using the concentration of measure techniques developed by Gromov and Milman, we prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology but not with the uniform one. Strengthening a de la Harpe's result, we show that a von Neumann algebra is approximately finite-dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group. To cite this article: T. Giordano, V. Pestov, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 273–278.     

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.     

Minimal self-joinings of infinite mixing actions of rank 1

We prove that measure-preserving actions of rank 1 of the groups ? ⁿ and ? ⁿ on a Lebesgue space with a σ-finite measure have minimal self-joinings.     

Geometry and dynamics of admissible metrics in measure spaces

We study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ?-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ?-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.     

Measure space automorphisms,the normalizers of their full groups,and approximate finiteness

We deal with the normalizer $\text{N}$[T] of the full group [T] of a nonsingular transformation T of a Lebesgue measure space in the group of all nonsingular transformations. We solve the conjugacy problem in $\text{N}$[T]/[T] for a measure preserving and ergodic T. Our results show that a locally finite extension of a solvable group is approximately finite.     

Transformations on [0, 1] with infinite invariant measures

Under certain regularity conditions a real transformation with indifferent fixed points has an infinite invariant measure equivalent to Lebesgue measure. In this paper several ergodic properties of such transformations are established.     

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.     

Weak rational ergodicity does not imply rational ergodicity

We construct an uncountable family of rank-one infinite measure-preserving transformations that are weakly rationally ergodic, but are not rationally ergodic, thus answering an open question by showing that weak rational ergodicity does not imply rational ergodicity.     

In some symmetric spaces monotonicity properties can be reduced to the cone of rearrangements

Geometric properties being the rearrangement counterparts of strict monotonicity, lower local uniform monotonicity and upper local uniform monotonicity in some symmetric spaces are considered. The relationships between strict monotonicity, upper local uniform monotonicity restricted to rearrangements and classical monotonicity properties (sometimes under some additional assumptions) are showed. It is proved that order continuity and lower uniform monotonicity properties for rearrangements of symmetric spaces together are equivalent to the classical lower local uniform monotonicity for any symmetric space over a \({\sigma}\)-finite complete and non-atomic measure space. It is also showed that in the case of order continuous symmetric spaces over a \({\sigma}\)-finite and complete measure space, upper local uniform monotonicity and its rearrangement counterpart shortly called ULUM* coincide. As an application of this result, in the case of a non-atomic complete finite measure a new proof of the theorem which is already known in the literature, giving the characterization of upper local uniform monotonicity of Orlicz–Lorentz spaces, is presented. Finally, it is proved that every rotund and reflexive space X such that both X and X* have the Kadec-Klee property is locally uniformly rotund. Some other results are also given in the first part of Sect. 2.     

Generators for amenable group actions

A generalisation of Krieger's finite generator theorem is proved for free actions of countable amenable groups on a non-atomic Lebesgue probability space.     

Mixing Sets for Rigid Transformations

Mathematical Notes - It is shown that, for any infinite set $$M\subset\mathbb N$$ of density zero, there exists a rigid measure-preserving transformation of a probability space which is mixing...     

Orbit equivalence rigidity for ergodic actions of the mapping class group

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher complexity. We prove similar rigidity results for a finite direct product of mapping class groups as well.      

Martingales,the smoothing index and ergodic theory

A classical theorem of Meyer Jerison which shows that the convergence in the pointwise ergodic theorem is equivalent to the convergence of an associated martingale is expanded to a conditional setting. An equiconvergence theorem of the type established for martingales by N.F.G. Martin and E. Boylan is established in the ergodic case for an ergodic, non-invertible, measure-preserving transformation.     

A generalization of Bihari’s lemma to the case of Volterra operators in Lebesgue spaces

For operators acting in the Lebesgue space L _q (Π), 1 < q < ∞, an abstract analog of Bihari’s lemma is stated and proved. We show that it can be used to derive a uniform pointwise estimate of the increment of the solution of a controlled functional-operator equation in the Lebesgue space. The procedure of reducing controlled initial boundary-value problems to this equation is illustrated by the Goursat-Darboux problem.     

Jointly ergodic measure-preserving transformations

The notion of ergodicity of a measure-preserving transformation is generalized to finite sets of transformations. The main result is that ifT ₁,T ₂, …,T _s are invertible commuting measure-preserving transformations of a probability space (X, ?, μ) then 1 $$\frac{1}{{N - M}}\sum\limits_{n = M}^{N - 1} {T{}_1^n } f_1 .T_2^n f_2 .....T_s^n f_s \xrightarrow[{N - M \to \propto }]{{I^2 (X)}}(\int_X {f1d\mu )} (\int_X {f2d\mu )...(\int_X {fsd\mu )} } $$ for anyf ₁,f ₂, …,f _s∈L ^x (X, ?, μ) iffT ₁×T ₂×…×T _s and all the transformationsT _iT_j ¹,i≠j, are ergodic. The multiple recurrence theorem for a weakly mixing transformation follows as a special case.