Strongly mixing sequences of measure preserving transformations

Ehrhard Behrends; Jörg Schmeling

Displaying similar documents to “Strongly mixing sequences of measure preserving transformations”

Mixing via families for measure preserving transformations

Rui Kuang, Xiangdong Ye (2008)

Colloquium Mathematicae

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In topological dynamics a theory of recurrence properties via (Furstenberg) families was established in the recent years. In the current paper we aim to establish a corresponding theory of ergodicity via families in measurable dynamical systems (MDS). For a family ℱ (of subsets of ℤ₊) and a MDS (X,,μ,T), several notions of ergodicity related to ℱ are introduced, and characterized via the weak topology in the induced Hilbert space L²(μ). T is ℱ-convergence ergodic of order k if for any...

Pointwise convergence for subsequences of weighted averages

Patrick LaVictoire (2011)

Colloquium Mathematicae

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We prove that if μₙ are probability measures on ℤ such that μ̂ₙ converges to 0 uniformly on every compact subset of (0,1), then there exists a subsequence $n_{k}$ such that the weighted ergodic averages corresponding to $μ_{n_{k}}$ satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n² + ⌊ρ(n)⌋ for a slowly growing function ρ. Under some monotonicity assumptions,...

Ergodic theorems in fully symmetric spaces of τ-measurable operators

Vladimir Chilin, Semyon Litvinov (2015)

Studia Mathematica

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Junge and Xu (2007), employing the technique of noncommutative interpolation, established a maximal ergodic theorem in noncommutative $L_{p}$ -spaces, 1 < p < ∞, and derived corresponding maximal ergodic inequalities and individual ergodic theorems. In this article, we derive maximal ergodic inequalities in noncommutative $L_{p}$ -spaces directly from the results of Yeadon (1977) and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these...

An anti-classification theorem for ergodic measure preserving transformations

Matthew Foreman, Benjamin Weiss (2004)

Journal of the European Mathematical Society

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Despite many notable advances the general problem of classifying ergodic measure preserving transformations (MPT) has remained wide open. We show that the action of the whole group of MPT’s on ergodic actions by conjugation is turbulent in the sense of G. Hjorth. The type of classifications ruled out by this property include countable algebraic objects such as those that occur in the Halmos–von Neumann theorem classifying ergodic MPT’s with pure point spectrum. We treat both the classical...

Pointwise ergodic theorems in Lorentz spaces L(p,q) for null preserving transformations

Ryotaro Sato (1995)

Studia Mathematica

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Let (X,ℱ,µ) be a finite measure space and τ a null preserving transformation on (X,ℱ,µ). Functions in Lorentz spaces L(p,q) associated with the measure μ are considered for pointwise ergodic theorems. Necessary and sufficient conditions are given in order that for any f in L(p,q) the ergodic average $n^{- 1} \sum_{i = 0}^{n - 1} f \circ τ^{i} (x)$ converges almost everywhere to a function f* in $L (p_{1}, q_{1}]$ , where (pq) and $(p_{1}, q_{1}]$ are assumed to be in the set $(r, s) : r = s = 1, o r 1 < r < \infty a n d 1 \leq s \leq \infty, o r r = s = \infty$ . Results due to C. Ryll-Nardzewski, S. Gładysz, and I. Assani and J. Woś are generalized...

Ergodic averages with deterministic weights

Fabien Durand, Dominique Schneider (2002)

Annales de l’institut Fourier

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We study the convergence of the ergodic averages $\frac{1}{N} \sum_{k = 0}^{N - 1} θ (k) f \circ T^{u_{k}}$ where ${(θ (k))}_{k \in ℕ}$ is a bounded sequence and ${(u_{k})}_{k \in ℕ}$ a strictly increasing sequence of integers such that ${Sup}_{α \in ℝ} | \sum_{k = 0}^{N - 1} θ (k) \exp (2 i π α u_{k}) | = O (N^{δ})$ for some $δ < 1$ . Moreover we give explicit such sequences $θ$ and $u$ and we investigate in particular the case where $θ$ is a $q$ -multiplicative sequence.

Weak almost periodicity of $L_{1}$ contractions and coboundaries of non-singular transformations

Isaac Kornfeld, Michael Lin (2000)

Studia Mathematica

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It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on $L_{1}$ is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex $L_{1}$ such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let...

Infinite measure preserving flows with infinite ergodic index

Alexandre I. Danilenko, Anton V. Solomko (2009)

Colloquium Mathematicae

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We construct a rank-one infinite measure preserving flow ${(T_{r})}_{r \in ℝ}$ such that for each p > 0, the “diagonal” flow ${(T_{r} \times \dots \times T_{r})}_{r \in ℝ} (p t i m e s)$ on the product space is ergodic.

Ergodic transforms associated to general averages

H. Aimar, A. L. Bernardis, F. J. Martín-Reyes (2010)

Studia Mathematica

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Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in $L^{p}$ , 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in $L^{p}$ . For p = 1 we find that the...

Ergodicity and conservativity of products of infinite transformations and their inverses

Julien Clancy, Rina Friedberg, Indraneel Kasmalkar, Isaac Loh, Tudor Pădurariu, Cesar E. Silva, Sahana Vasudevan (2016)

Colloquium Mathematicae

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We construct a class of rank-one infinite measure-preserving transformations such that for each transformation T in the class, the cartesian product T × T is ergodic, but the product $T \times T^{- 1}$ is not. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.

A probabilistic ergodic decomposition result

Albert Raugi (2009)

Annales de l'I.H.P. Probabilités et statistiques

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Let $(X, 𝔛, μ)$ be a standard probability space. We say that a sub--algebra $𝔅$ of $𝔛$ if any regular conditional probability $^{𝔅} P$ with respect to $𝔅$ and satisfies, for -almost every ∈, $\forall B \in 𝔅,^{𝔅} P (x, B) \in {0, 1}$ . In this case the equality $μ (\cdot) = \int_{X}^{𝔅} P (x, \cdot) μ (d x)$ , gives us an integral decomposition in “ $𝔅$ -ergodic” components. For any sub--algebra $𝔅$ of $𝔛$ , we denote by $\bar{𝔅}$ the smallest sub--algebra of $𝔛$ containing $𝔅$ and the collection of all setsin $𝔛$ satisfying()=0. We say that $𝔅$ is-complete if $𝔅 = \bar{𝔅}$ . Let ${𝔅_{i} i \in I}$ be a non-empty family of sub--algebras...

JOP's counting function and Jones' square function

Karin Reinhold (2006)

Studia Mathematica

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We study a class of square functions in a general framework with applications to a variety of situations: samples along subsequences, averages of $ℤ ₊^{d}$ actions and of positive L¹ contractions. We also study the relationship between a counting function first introduced by Jamison, Orey and Pruitt, in a variety of situations, and the corresponding ergodic averages. We show that the maximal counting function is not dominated by the square functions.

On the sequence of integer parts of a good sequence for the ergodic theorem

Emmanuel Lesigne (1995)

Commentationes Mathematicae Universitatis Carolinae

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If $(u_{n})$ is a sequence of real numbers which is good for the ergodic theorem, is the sequence of the integer parts $([u_{n}])$ good for the ergodic theorem ? The answer is negative for the mean ergodic theorem and affirmative for the pointwise ergodic theorem.