Ergodic averages with deterministic weights

Fabien Durand; Dominique Schneider

Displaying similar documents to “Ergodic averages with deterministic weights”

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,...

Strongly mixing sequences of measure preserving transformations

Ehrhard Behrends, Jörg Schmeling (2001)

Czechoslovak Mathematical Journal

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We call a sequence $(T_{n})$ of measure preserving transformations strongly mixing if $P (T_{n}^{- 1} A \cap B)$ tends to $P (A) P (B)$ for arbitrary measurable $A$ , $B$ . We investigate whether one can pass to a suitable subsequence $(T_{n_{k}})$ such that $\frac{1}{K} \sum_{k = 1}^{K} f (T_{n_{k}}) ⟶ \int f d P$ almost surely for all (or “many”) integrable $f$ .

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...

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...

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.

Multiparameter admissible superadditive processes

Doğan Çömez (2005)

Colloquium Mathematicae

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In this article some properties of Markovian mean ergodic operators are studied. As an application of the tools developed, and using the admissibility feature, a “reduction of order” technique for multiparameter admissible superadditive processes is obtained. This technique is later utilized to obtain a.e. convergence of averages $n^{- 2} \sum_{i, j = 0}^{n - 1} f_{(i, j)}$ as well as their weighted version.

Spectral decompositions, ergodic averages, and the Hilbert transform

Earl Berkson, T. A. Gillespie (2001)

Studia Mathematica

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Let U be a trigonometrically well-bounded operator on a Banach space , and denote by ${ₙ (U)}_{n = 1}^{\infty}$ the sequence of (C,2) weighted discrete ergodic averages of U, that is, $ₙ (U) = 1 / n \sum_{0 < | k | \leq n} (1 - | k | / (n + 1)) U^{k}$ . We show that this sequence ${ₙ (U)}_{n = 1}^{\infty}$ of weighted ergodic averages converges in the strong operator topology to an idempotent operator whose range is x ∈ : Ux = x, and whose null space is the closure of (I - U). This result expands the scope of the traditional Ergodic Theorem, and thereby serves as a link between Banach space spectral...

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...

Hamiltonian loops from the ergodic point of view

Leonid Polterovich (1999)

Journal of the European Mathematical Society

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Let $G$ be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold $Y$ . A loop $h : S^{1} \to G$ is called strictly ergodic if for some irrational number the associated skew product map $T : S^{1} \times Y \to S^{1} \times Y$ defined by $T (t, y) = (t + α; h (t) y)$ is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of $G$ can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply...

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.

Almost everywhere convergence of generalized ergodic transforms for invertible power-bounded operators in $L^{p}$

Christophe Cuny (2011)

Colloquium Mathematicae

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We show that some results of Gaposhkin about a.e. convergence of series associated to a unitary operator U acting on L²(X,Σ,μ) (μ is a σ-finite measure) may be extended to the case where U is an invertible power-bounded operator acting on $L^{p} (X, Σ, μ)$ , p > 1. The proofs make use of the spectral integration initiated by Berkson-Gillespie and, more specifically, of recent results of the author.

Positive L¹ operators associated with nonsingular mappings and an example of E. Hille

Isaac Kornfeld, Wojciech Kosek (2003)

Colloquium Mathematicae

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E. Hille [Hi1] gave an example of an operator in L¹[0,1] satisfying the mean ergodic theorem (MET) and such that supₙ||Tⁿ|| = ∞ (actually, $| | T ⁿ | | \sim n^{1 / 4}$ ). This was the first example of a non-power bounded mean ergodic L¹ operator. In this note, the possible rates of growth (in n) of the norms of Tⁿ for such operators are studied. We show that, for every γ > 0, there are positive L¹ operators T satisfying the MET with $l i m_{n \to \infty} | | T ⁿ | | / n^{1 - γ} = \infty . I n t h e c l a s s o f p o s i t i v e o p e r a t o r s t h e s e e x a m p l e s a r e t h e b e s t p o s s i b l e i n t h e s e n s e t h a t f o r e v e r y s u c h o p e r a t o r T t h e r e e x i s t s a γ ₀ > 0 s u c h t h a t$ lim supn→ ∞ ||Tⁿ||/n1-γ₀ = 0 $.$ A class of numerical sequences αₙ, intimately...

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...

Spectral theory and operator ergodic theory on super-reflexive Banach spaces

Earl Berkson (2010)

Studia Mathematica

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On reflexive spaces trigonometrically well-bounded operators have an operator-ergodic-theory characterization as the invertible operators U such that $s u p_{n \in ℕ, z \in} | | \sum_{0 < | k | \leq n} (1 - | k | / (n + 1)) k^{- 1} z^{k} U^{k} | | < \infty$ . (*) Trigonometrically well-bounded operators permeate many settings of modern analysis, and this note highlights the advances in both their spectral theory and operator ergodic theory made possible by a recent rekindling of interest in the R. C. James inequalities for super-reflexive spaces. When the James inequalities are combined with...

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.