We prove that for a generic measure preserving transformation T , the closed group generated by T is a continuous homomorphic image of a closed linear subspace of L0(λ,R), where λ is Lebesgue measure, and that the closed group generated by T contains an increasing sequence of finite dimensional toruses whose union is dense.

We prove that the notions of Krengel entropy and Poisson entropy for infinite-measure-preserving transformations do not always coincide: We construct a conservative infinite-measure-preserving transformation with zero Krengel entropy (the induced transformation on a set of measure 1 is the Von Neumann–Kakutani odometer), but whose associated Poisson suspension has positive entropy. Résumé. Nous...

For a σ-finite measure preserving dynamical system (X,μ, T ), we formulate necessary and sufficient conditions for a Young tower (∆, ν, F ) to be a (measure theoretic) extension of the original system. Because F is pointwise dual ergodic by construction, one immediate consequence of these conditions is that the Darling Kac theorem carries over from F to T . One advantage of the Darling Kac theo...

Let T1, . . . , Tl : X → X be commuting measure-preserving transformations on a probability space (X,X , μ). We show that the multiple ergodic averages 1 N PN−1 n=0 f1(T n 1 x) . . . fl(T n l x) are convergent in L2(X,X , μ) as N → ∞ for all f1, . . . , fl ∈ L (X,X , μ); this was previously established for l = 2 by Conze and Lesigne [2] and for general l assuming some additional ergodicity hypo...

Let α be a Zd–action (d ≥ 2) by automorphisms of a compact metric abelian group. For any non–linear shape I ⊂ Zd, there is an α with the property that I is a minimal mixing shape for α. The only implications of the form “I is a mixing shape for α =⇒ J is a mixing shape for α” are trivial ones for which I contains a translate of J . If all shapes are mixing for α, then α is mixing of all orders....

Almost every, essentially: Given a Lebesgue measure space (X,B, μ), a property P (x) predicated of elements of X is said to hold for almost every x ∈ X, if the set X \ {x : P (x) holds} has zero measure. Two sets A,B ∈ B are essentially disjoint if μ(A ∩B) = 0. Conservative system: Is an infinite measure preserving system such that for no set A ∈ B with positive measure are A,T−1A,T−2A, . . . p...

Partial results were obtained previously by K. F. Roth (1952) who established the existence of arithmetic progressions of length three in subsets of Z of positive upper density, and by E. Szemerédi (1969) who proved the existence of progressions of length four. In 1976 Furstenberg noticed that the statement of Theorem I is equivalent to a statement about "multiple recurrence" of measure-preserv...

In this paper, we demonstrate how approximation structures called sufficient semirings can provide information about measure-preserving dynamical systems. We describe the basic properties of measure-preserving dynamical systems and illustrate their connections to sufficient semirings by investigating the dyadic adding machine.

We define a theory of noncommutative general relativity for canonical noncommutative spaces. We find a subclass of general coordinate transformations acting on canonical noncommutative spacetimes to be volume-preserving transformations. Local Lorentz invariance is treated as a gauge theory with the spin connection field taken in the so(3,1) enveloping algebra. The resulting theory appears to be...

converge almost everywhere to a finite limit f*(x). It then follows that the limit function/* is integrable and that/*(7x) =/*(x) almost everywhere. This result can be applied to certain cases in which the given measure m is not preserved by the transformation T. In order to discuss this application, we recall some terminology for measures and transformations. If (X, S) is a measurable space, a...