Almost uniform convergence in the Wiener–Wintner ergodic theorem (original) (raw)

Almost uniform convergence in Wiener-Wintner ergodic theorem

arXiv (Cornell University), 2020

We extend almost everywhere convergence in Wiener-Wintner ergodic theorem for σ-finite measure to a generally stronger almost uniform convergence and present a larger, universal, space for which this convergence holds. We then extend this result to the case with Besicovitch weights. n−1 k=0 f • T k. Definition 1.1. We write f ∈ a.e. W W (Ω, T) (f ∈ a.u. W W (Ω, T)) if ∃ Ω f ⊂ Ω with µ(Ω \ Ω f) = 0 such that the sequence M n (T, λ)(f)(ω) converges for any ω ∈ Ω f and λ ∈ C 1. (respectively, if ∀ ε > 0 ∃ Ω ′ = Ω f,ε with µ(Ω\Ω ′) ≤ ε such that the sequence M n (T, λ)(f)χ Ω ′

On the almost everywhere convergence of the ergodic averages

Ergodic Theory and Dynamical Systems, 1990

Let (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).

Almost uniform and strong convergences in ergodic theorems for symmetric spaces

Acta Mathematica Hungarica

Let (Ω, µ) be a σ-finite measure space, and let X ⊂ L 1 (Ω)+L ∞ (Ω) be a fully symmetric space of measurable functions on (Ω, µ). If µ(Ω) = ∞, necessary and sufficient conditions are given for almost uniform convergence in X (in Egorov's sense) of Cesàro averages Mn(T)(f) = 1 n n−1 k=0 T k (f) for all Dunford-Schwartz operators T in L 1 (Ω) + L ∞ (Ω) and any f ∈ X. Besides, it is proved that the averages Mn(T) converge strongly in X for each Dunford-Schwartz operator T in L 1 (Ω) + L ∞ (Ω) if and only if X has order continuous norm and L 1 (Ω) is not contained in X.

(Uniform) convergence of twisted ergodic averages

Ergodic Theory and Dynamical Systems, 2015

Let T be an ergodic measure-preserving transformation on a nonatomic probability space (X, Σ, µ). We prove uniform extensions of the Wiener-Wintner theorem in two settings:

Random ergodic theorems and regularizing random weights

Ergodic Theory and Dynamical Systems, 2003

We study the convergence of pointwise ergodic means for random subsequences, in a universal framework, together with ergodic means which are modulated by random weights. The methods used in this work mainly involve Gaussian tools, transference principles and new results on oscillation functions.

On the Wiener-Wintner theorem for nilsequences

2012

We give a Fourier analytic proof of the generalisation due to Host and Kra of the classical Wiener-Wintner theorem and give some explicit bounds on the limit of the weighted ergodic averages.

On pointwise ergodic theorems for infinite measure

arXiv (Cornell University), 2015

For a Dunford-Schwartz operator in the L p −space, 1 ≤ p < ∞, of an arbitrary measure space, we prove pointwise convergence of the conventional and Besicovitch weighted ergodic averages. Pointwise convergence of various types of ergodic averages in fully symmetric spaces of measurable functions with non-trivial Boyd indices is studied. In particular, it is shown that for such spaces Bourgain's Return Times theorem is valid. Definition 1.1. A measure space (Ω, A, µ) is called semifinite if every subset of Ω of non-zero measure admits a subset of finite non-zero measure. A semifinite measure space (Ω, A, µ) is said to have the direct sum property if the Boolean algebra (A/ ∼) of equivalence classes of measurable sets is complete, that is, every subset of (A/ ∼) has a least upper bound. Note that every σ−finite measure space has the direct sum property. A detailed account on measures with direct sum property is found in [5]; see also [12].

Individual ergodic theorems for infinite measure

Colloquium Mathematicum

Given a σ-finite infinite measure space (Ω, µ), it is shown that any Dunford-Schwartz operator T : L 1 (Ω) → L 1 (Ω) can be uniquely extended to the space L 1 (Ω) + L ∞ (Ω). This allows to find the largest subspace Rµ of L 1 (Ω) + L ∞ (Ω) such that the ergodic averages 1 n n−1 k=0 T k (f) converge almost uniformly (in Egorov's sense) for every f ∈ Rµ and every Dunford-Schwartz operator T. Utilizing this result, almost uniform convergence of the averages 1 n n−1 k=0 β k T k (f) for every f ∈ Rµ, any Dunford-Schwartz operator T and any bounded Besicovitch sequence {β k } is established. Further, given a measure preserving transformation τ : Ω → Ω, Assani's extension of Bourgain's Return Times theorem to σ-finite measure is employed to show that for each f ∈ Rµ there exists a set Ω f ⊂ Ω such that µ(Ω \ Ω f) = 0 and the averages 1 n n−1 k=0 β k f (τ k ω) converge for all ω ∈ Ω f and any bounded Besicovitch sequence {β k }. Applications to fully symmetric subspaces E ⊂ Rµ are given.

Ergodic averages with generalized weights

Studia Mathematica, 2006

Two types of weighted ergodic averages are studied. It is shown that if F = {F n } is an admissible superadditive process relative to a measure preserving transformation, then a Wiener-Wintner type result holds for F. Using this result new good classes of weights generated by such processes are obtained. We also introduce another class of weights via the group of unitary functions, and study the convergence of the corresponding weighted averages. The limits of such weighted averages are also identified.