scholarly journals (Uniform) convergence of twisted ergodic averages

2015 ◽  
Vol 36 (7) ◽  
pp. 2172-2202 ◽  
Author(s):  
TANJA EISNER ◽  
BEN KRAUSE
Keyword(s):  
Uniform Convergence ◽  
Probability Space ◽  
Elementary Proof ◽  
Ergodic Measure ◽  
Ergodic Averages ◽  
Almost Everywhere ◽  
Measure Preserving Transformation ◽  
Image Position ◽  
Badly Approximable

Let$T$be an ergodic measure-preserving transformation on a non-atomic probability space$(X,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$. We prove uniform extensions of the Wiener–Wintner theorem in two settings: for averages involving weights coming from Hardy field functions $p$,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(p(n))T^{n}f(x)\bigg\}; & & \displaystyle \nonumber\end{eqnarray}$$and for ‘twisted’ polynomial ergodic averages,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(n\unicode[STIX]{x1D703})T^{P(n)}f(x)\bigg\} & & \displaystyle \nonumber\end{eqnarray}$$for certain classes of badly approximable$\unicode[STIX]{x1D703}\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise$\unicode[STIX]{x1D707}$-almost everywhere for$f\in L^{p}(X),p>1,$and arbitrary$\unicode[STIX]{x1D703}\in [0,1]$.

Weak-type inequalities and convergence of the ergodic averages in variable Lebesgue spaces with weights

2009 ◽  
Vol 139 (4) ◽  
pp. 673-683 ◽  
Author(s):  
M. Isabel Aguilar Cañestro ◽  
Pedro Ortega Salvador
Keyword(s):  
Lebesgue Space ◽  
Weak Type ◽  
Suitable Condition ◽  
Variable Exponents ◽  
Ergodic Averages ◽  
Lebesgue Spaces ◽  
Variable Lebesgue Spaces ◽  
Almost Everywhere ◽  
Variable Lebesgue Space ◽  
Measure Preserving Transformation

We characterize the weighted weak-type inequalities with variable exponents for the maximal operator associated with an ergodic, invertible, measure-preserving transformation and prove the almost everywhere convergence of the ergodic averages for all functions in a variable Lebesgue space with a weight verifying a suitable condition.

The Pointwise Ergodic Theorem for Transformations whose Orbits contain or are contained in the Orbits of a Measure-Preserving Transformation

1982 ◽  
Vol 34 (6) ◽  
pp. 1303-1318 ◽  
Author(s):  
John C. Kieffer ◽  
Maurice Rahe
Keyword(s):  
Information Theory ◽  
Ergodic Theory ◽  
Probability Space ◽  
Sufficient Conditions ◽  
Measurable Transformation ◽  
Input And Output ◽  
Measure Preserving Transformation ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Measure Preserving Transformations

1. Introduction. Let be a probability space with standard. Let T be a bimeasurable one-to-one map of Ω onto itself. Let U: Ω → Ω be another measurable transformation whose orbits are contained in the T-orbits; that is,where Z denotes the set of integers. (This is equivalent to saying that there is a measurable mapping L: Ω → Z such that U(ω) = TL(ω)(ω), ω ∈ Ω.) Such pairs (T, U) arise quite naturally in ergodic theory and information theory. (For example, in ergodic theory, one can see such pairs in the study of the full group of a transformation [1]; in information theory, such a pair can be associated with the input and output of a variable-length source encoder [2] [3].) Neveu [4] obtained necessary and sufficient conditions for U to be measure-preserving if T is measure-preserving. However, if U fails to be measure-preserving, U might still possess many of the features of measure-preserving transformations.

Ergodic Averages for Weight Functions Moved by Non-Linear Transformations on ℝn

1995 ◽  
Vol 47 (4) ◽  
pp. 852-876
Author(s):  
David I. McIntosh
Keyword(s):  
Sufficient Conditions ◽  
Convergent Sequence ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Linear Transformations ◽  
Borel Sets ◽  
Almost Everywhere ◽  
Image Position ◽  
Bounded Sets

AbstractLet ℝ+ denote the non-negative half of the real line, and let λ denote Lebesgue measure on the Borel sets of ℝn. A function φ: ℝn → ℝ+ is called a weight function if ʃℝn φ dλ = 1. Let (X, ℱ, μ) be a non-atomic, finite measure space, let ƒ: X → ℝ+, and suppose { Tν}ν∊ℝn is an ergodic, aperiodic ℝn-flow on X. We consider the weighted ergodic averages where is a sequence of weight functions. Sufficient as well as necessary and sufficient conditions for the pointwise, almost-everywhere convergence of are developed for a particular class of weight functions φk. Specifically, let {τk: ℝn → ℝn} be a sequence of measurable, non-singular maps with measurable, non-singular inverses such that the Radon-Nikodym derivatives dλ oτk /dλ and dλ oτk-1 / dλ are L∞ (ℝn), and such that τk and τ-1 map bounded sets to bounded sets. We examine convergence for the sequence where θk is an a.e.-convergent sequence of weight functions which are dominated by a fixed L1(ℝn) function with bounded support.

scholarly journals Entropy of systems

2016 ◽  
Vol 38 (3) ◽  
pp. 1118-1126
Author(s):  
RADU B. MUNTEANU
Keyword(s):  
Positive Integer ◽  
Probability Space ◽  
Bernoulli Shift ◽  
Ergodic Measure ◽  
Measure Preserving Transformation ◽  
Zero Entropy ◽  
Standard Probability

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is $\text{AT}(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not $\text{AT}(n)$. We also give an example of a transformation which has zero entropy but does not have property $\text{AT}(n)$ for any integer $n\geq 1$.

An elementary proof of the Birkhoff-Hopf theorem

1995 ◽  
Vol 117 (1) ◽  
pp. 31-55 ◽  
Author(s):  
Simon P. Eveson ◽  
Roger D. Nussbaum
Keyword(s):  
Elementary Proof ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Important Work ◽  
Large Classes ◽  
Contraction Mappings ◽  
Measurable Functions ◽  
Almost Everywhere ◽  
Image Position ◽  
Hopf Theorem

In important work some thirty years ago, G. Birkhoff[2, 3] and E. Hopf [16, 17] showed that large classes of positive linear operators behave like contraction mappings with respect to certain ‘almost’ metrics. Hopf worked in a space of measurable functions and took as his ‘almost’ metric the oscillation ω(y/x) of functions y and x with x(t) > 0 almost everywhere, defined by

scholarly journals Weak ergodic averages over dilated measures

2019 ◽  
Vol 41 (2) ◽  
pp. 606-621
Author(s):  
WENBO SUN
Keyword(s):  
Harmonic Analysis ◽  
Pointwise Convergence ◽  
Borel Measure ◽  
Ergodic Measure ◽  
Ergodic Average ◽  
Theoretic Approach ◽  
Ergodic System ◽  
Ergodic Averages ◽  
Image Position

Let $m\in \mathbb{N}$ and $\mathbf{X}=(X,{\mathcal{X}},\unicode[STIX]{x1D707},(T_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}})$ be a measure-preserving system with an $\mathbb{R}^{m}$-action. We say that a Borel measure $\unicode[STIX]{x1D708}$ on $\mathbb{R}^{m}$ is weakly equidistributed for $\mathbf{X}$ if there exists $A\subseteq \mathbb{R}$ of density 1 such that, for all $f\in L^{\infty }(\unicode[STIX]{x1D707})$, we have $$\begin{eqnarray}\lim _{t\in A,t\rightarrow \infty }\int _{\mathbb{R}^{m}}f(T_{t\unicode[STIX]{x1D6FC}}x)\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FC})=\int _{X}f\,d\unicode[STIX]{x1D707}\end{eqnarray}$$ for $\unicode[STIX]{x1D707}$-almost every $x\in X$. Let $W(\mathbf{X})$ denote the collection of all $\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}$ such that the $\mathbb{R}$-action $(T_{t\unicode[STIX]{x1D6FC}})_{t\in \mathbb{R}}$ is not ergodic. Under the assumption of the pointwise convergence of the double Birkhoff ergodic average, we show that a Borel measure $\unicode[STIX]{x1D708}$ on $\mathbb{R}^{m}$ is weakly equidistributed for an ergodic system $\mathbf{X}$ if and only if $\unicode[STIX]{x1D708}(W(\mathbf{X})+\unicode[STIX]{x1D6FD})=0$ for every $\unicode[STIX]{x1D6FD}\in \mathbb{R}^{m}$. Under the same assumption, we also show that $\unicode[STIX]{x1D708}$ is weakly equidistributed for all ergodic measure-preserving systems with $\mathbb{R}^{m}$-actions if and only if $\unicode[STIX]{x1D708}(\ell )=0$ for all hyperplanes $\ell$ of $\mathbb{R}^{m}$. Unlike many equidistribution results in literature whose proofs use methods from harmonic analysis, our results adopt a purely ergodic-theoretic approach.

Almost Everywhere Convergence of Convolution Measures

2012 ◽  
Vol 55 (4) ◽  
pp. 830-841
Author(s):  
Karin Reinhold ◽  
Anna K. Savvopoulou ◽  
Christopher M. Wedrychowicz
Keyword(s):  
Dynamical System ◽  
Probability Space ◽  
Probability Measures ◽  
Almost Everywhere Convergence ◽  
Weighted Averages ◽  
Almost Everywhere ◽  
Measure Preserving Transformation ◽  
Convolution Products ◽  
Maximal Estimate ◽  
Dense Class

AbstractLet (X, ℬ, m, τ) be a dynamical system with (X, ℬ, m) a probability space and τ an invertible, measure preserving transformation. This paper deals with the almost everywhere convergence in L1(X) of a sequence of operators of weighted averages. Almost everywhere convergence follows once we obtain an appropriate maximal estimate and once we provide a dense class where convergence holds almost everywhere. The weights are given by convolution products of members of a sequence of probability measures {vi} defined on ℤ. We then exhibit cases of such averages where convergence fails.

scholarly journals Convergence for moving averages

1990 ◽  
Vol 10 (1) ◽  
pp. 43-62 ◽  
Author(s):  
Alexandra Bellow ◽  
Roger Jones ◽  
Joseph Rosenblatt
Keyword(s):  
Probability Space ◽  
Sufficient Conditions ◽  
Ergodic Measure ◽  
Point Transformation ◽  
Necessary And Sufficient Conditions ◽  
Moving Averages ◽  
Averaging Operators ◽  
Almost Everywhere ◽  
Necessary And Sufficient ◽  
Positive Integers

AbstractAssume T is an ergodic measure preserving point transformation from a probability space onto itself. Let be a sequence of pairs of positive integers, and define the sequence of averaging operators . Necessary and sufficient conditions are given forthis sequence of averages to converge almost everywhere. Weighted versions are also considered.

scholarly journals Pointwise convergence of certain continuous-time double ergodic averages

2021 ◽  
pp. 1-11
Author(s):  
MICHAEL CHRIST ◽  
POLONA DURCIK ◽  
VJEKOSLAV KOVAČ ◽  
JORIS ROOS
Keyword(s):  
Recent Work ◽  
Hilbert Transform ◽  
Continuous Time ◽  
Pointwise Convergence ◽  
Probability Space ◽  
Singular Integrals ◽  
Ergodic Averages ◽  
Almost Everywhere Convergence ◽  
Almost Everywhere ◽  
Multilinear Singular Integrals

Abstract We prove almost everywhere convergence of continuous-time quadratic averages with respect to two commuting $\mathbb {R}$ -actions, coming from a single jointly measurable measure-preserving $\mathbb {R}^2$ -action on a probability space. The key ingredient of the proof comes from recent work on multilinear singular integrals; more specifically, from the study of a curved model for the triangular Hilbert transform.

scholarly journals Convergence of ergodic averages for many group rotations

2015 ◽  
Vol 36 (7) ◽  
pp. 2107-2120
Author(s):  
ZOLTÁN BUCZOLICH ◽  
GABRIELLA KESZTHELYI
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Measurable Function ◽  
Dual Group ◽  
Ergodic Averages ◽  
Abelian Topological Group ◽  
Compact Abelian Topological Group ◽  
Image Position ◽  
Rotation Set ◽  
Group Rotations

Suppose that $G$ is a compact Abelian topological group, $m$ is the Haar measure on $G$ and $f:G\rightarrow \mathbb{R}$ is a measurable function. Given $(n_{k})$, a strictly monotone increasing sequence of integers, we consider the non-conventional ergodic/Birkhoff averages $$\begin{eqnarray}M_{N}^{\unicode[STIX]{x1D6FC}}f(x)=\frac{1}{N+1}\mathop{\sum }_{k=0}^{N}f(x+n_{k}\unicode[STIX]{x1D6FC}).\end{eqnarray}$$ The $f$-rotation set is $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{f}=\{\unicode[STIX]{x1D6FC}\in G:M_{N}^{\unicode[STIX]{x1D6FC}}f(x)\text{ converges for }m\text{ almost every }x\text{ as }N\rightarrow \infty \}.\end{eqnarray}$$We prove that if $G$ is a compact locally connected Abelian group and $f:G\rightarrow \mathbb{R}$ is a measurable function then from $m(\unicode[STIX]{x1D6E4}_{f})>0$ it follows that $f\in L^{1}(G)$. A similar result is established for ordinary Birkhoff averages if $G=Z_{p}$, the group of $p$-adic integers. However, if the dual group, $\widehat{G}$, contains ‘infinitely many multiple torsion’ then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, $f(x+n_{k}\unicode[STIX]{x1D6FC})/k$, $k=1,\ldots ,$ for almost every $x$ for many $\unicode[STIX]{x1D6FC}$; hence, some of our theorems are stated by using instead of $\unicode[STIX]{x1D6E4}_{f}$ slightly larger sets, denoted by $\unicode[STIX]{x1D6E4}_{f,b}$.

Sign in / Sign up

Export Citation Format

Share Document