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.

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$\unicode[STIX]{x1D703}\in [0,1]$.

scholarly journals Pointwise multiple averages for sublinear functions

2018 ◽  
Vol 40 (6) ◽  
pp. 1594-1618
Author(s):  
SEBASTIÁN DONOSO ◽  
ANDREAS KOUTSOGIANNIS ◽  
WENBO SUN
Keyword(s):  
Large Class ◽  
Explicit Formula ◽  
Pointwise Convergence ◽  
Limit Function ◽  
Invariant Property ◽  
Ergodic Averages ◽  
Order Zero ◽  
Good For ◽  
Sublinear Functions ◽  
Uniquely Ergodic

For any measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T_{1},\ldots ,T_{d})$ with no commutativity assumptions on the transformations $T_{i},$$1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of different growth coming from a large class of sublinear functions. This class properly contains important subclasses of Hardy field functions of order zero and of Fejér functions, i.e., tempered functions of order zero. We show that the convergence of the single average, via an invariant property, implies the convergence of the multiple one. We also provide examples of sublinear functions which are, in general, bad for convergence on arbitrary systems, but good for uniquely ergodic systems. The case where the fastest function is linear is addressed as well, and we provide, in all the cases, an explicit formula of the limit function.

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 $\unicode[STIX]{x1D6E4}_{f,b}$.

scholarly journals Ergodic averages with prime divisor weights in

2017 ◽  
Vol 39 (4) ◽  
pp. 889-897 ◽  
Author(s):  
ZOLTÁN BUCZOLICH
Keyword(s):  
Dynamical System ◽  
Ergodic Theorem ◽  
Prime Divisor ◽  
Ergodic Averages ◽  
Weighting Functions ◽  
Pointwise Ergodic Theorem ◽  
Prime Factors ◽  
Ergodic Dynamical System ◽  
Image Position

We show that $\unicode[STIX]{x1D714}(n)$ and $\unicode[STIX]{x1D6FA}(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity, are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\sum _{n\leq K}g(n)$, then for every ergodic dynamical system $(X,{\mathcal{A}},\unicode[STIX]{x1D707},\unicode[STIX]{x1D70F})$ and every $f\in L^{1}(X)$, $$\begin{eqnarray}\lim _{K\rightarrow \infty }\frac{1}{S_{g,K}}\mathop{\sum }_{n=1}^{K}g(n)f(\unicode[STIX]{x1D70F}^{n}x)=\int _{X}f\,d\unicode[STIX]{x1D707}\quad \text{for }\unicode[STIX]{x1D707}\text{ almost every }x\in X.\end{eqnarray}$$ This answers a question raised by Cuny and Weber, who showed this result for $L^{p}$, $p>1$.

Moving Weighted Averages

1993 ◽  
Vol 45 (3) ◽  
pp. 449-469 ◽  
Author(s):  
M. A. Akcoglu ◽  
Y. Déniel
Keyword(s):  
Weight Function ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

scholarly journals Radial growth and exceptional sets for Cauchy–Stieltjes integrals

1994 ◽  
Vol 37 (1) ◽  
pp. 73-89 ◽  
Author(s):  
D. J. Hallenbeck ◽  
T. H. MacGregor
Keyword(s):  
Unit Circle ◽  
Radial Growth ◽  
Borel Measure ◽  
Local Conditions ◽  
Exceptional Sets ◽  
Image Position ◽  
Complex Valued

This paper considers the radial and nontangential growth of a function f given bywhere α>0 and μ is a complex-valued Borel measure on the unit circle. The main theorem shows how certain local conditions on μ near eiθ affect the growth of f(z) as z→eiθ in Stolz angles. This result leads to estimates on the nontangential growth of f where exceptional sets occur having zero β-capacity.

scholarly journals On Theorems of Kawada and Wendel

1958 ◽  
Vol 11 (2) ◽  
pp. 71-77 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Convolution Product ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Complex Functions ◽  
Image Position ◽  
Locally Compact Topological Group ◽  
Invariant Haar Measure

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li byis again in L1, and

scholarly journals Norm variation of ergodic averages with respect to two commuting transformations

2017 ◽  
Vol 39 (3) ◽  
pp. 658-688 ◽  
Author(s):  
POLONA DURCIK ◽  
VJEKOSLAV KOVAČ ◽  
KRISTINA ANA ŠKREB ◽  
CHRISTOPH THIELE
Keyword(s):  
Harmonic Analysis ◽  
Hilbert Transform ◽  
Singular Integrals ◽  
Quantitative Result ◽  
Square Function ◽  
Two Dimensional ◽  
Ergodic Averages ◽  
Multilinear Singular Integrals

We study double ergodic averages with respect to two general commuting transformations and establish a sharp quantitative result on their convergence in the norm. We approach the problem via real harmonic analysis, using recently developed methods for bounding multilinear singular integrals with certain entangled structure. A byproduct of our proof is a bound for a two-dimensional bilinear square function related to the so-called triangular Hilbert transform.

scholarly journals Pointwise characteristic factors for Wiener–Wintner double recurrence theorem

2015 ◽  
Vol 36 (4) ◽  
pp. 1037-1066 ◽  
Author(s):  
IDRIS ASSANI ◽  
DAVID DUNCAN ◽  
RYO MOORE
Keyword(s):  
Full Measure ◽  
Orthogonal Complement ◽  
Ergodic System ◽  
Absolute Value ◽  
Recurrence Theorem ◽  
The Absolute ◽  
Image Position ◽  
Null Set

In this paper we extend Bourgain’s double recurrence result to the Wiener–Wintner averages. Let $(X,{\mathcal{F}},{\it\mu},T)$ be a standard ergodic system. We will show that for any $f_{1},f_{2}\in L^{\infty }(X)$, the double recurrence Wiener–Wintner average $$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)e^{2{\it\pi}int}\end{eqnarray}$$ converges off a single null set of $X$ independent of $t$ as $N\rightarrow \infty$. Furthermore, we will show a uniform Wiener–Wintner double recurrence result: if either $f_{1}$ or $f_{2}$ belongs to the orthogonal complement of the Conze–Lesigne factor, then there exists a set of full measure such that the supremum on $t$ of the absolute value of the averages above converges to $0$.

Sign in / Sign up

Export Citation Format

Share Document