scholarly journals Random sequences and pointwise convergence of multiple ergodic averages

2012 ◽  
Vol 61 (2) ◽  
pp. 585-617 ◽  
Author(s):  
Nikos Frantzikinakis ◽  
Emmanuel Lesigne ◽  
Máté Wierdl
Keyword(s):  
Pointwise Convergence ◽  
Ergodic Averages ◽  
Random Sequences

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.

Pointwise convergence for cubic and polynomial multiple ergodic averages of non-commuting transformations

2011 ◽  
Vol 32 (3) ◽  
pp. 877-897 ◽  
Author(s):  
QING CHU ◽  
NIKOS FRANTZIKINAKIS
Keyword(s):  
Ergodic Theory ◽  
Pointwise Convergence ◽  
Ergodic Averages ◽  
Limiting Behavior ◽  
Measure Preserving Transformations ◽  
Bounded Sequences

AbstractWe study the limiting behavior of multiple ergodic averages involving several, not necessarily commuting, measure-preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and another that uses iterates along shifted polynomials. We prove pointwise convergence in both cases, thus answering a question of I. Assani in the former case, and extending the results of B. Host and B. Kra, and A. Leibman in the latter case. Our argument is based on some elementary uniformity estimates of general bounded sequences, decomposition results in ergodic theory, and equidistribution results on nilmanifolds.

scholarly journals Powers of sequences and convergence of ergodic averages

2009 ◽  
Vol 30 (5) ◽  
pp. 1431-1456 ◽  
Author(s):  
N. FRANTZIKINAKIS ◽  
M. JOHNSON ◽  
E. LESIGNE ◽  
M. WIERDL
Keyword(s):  
Ergodic Theorem ◽  
Measurable Function ◽  
Pointwise Convergence ◽  
Ergodic Averages ◽  
Bounded Measurable Function ◽  
Mean Ergodic Theorem ◽  
The Mean ◽  
Good For

AbstractA sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure-preserving system (X,ℬ,μ,T) and any bounded measurable function f, the averages (1/N)∑ Nn=1f(Tsnx) converge in the L2(μ) norm. We construct a sequence (sn) which is good for the mean ergodic theorem but such that the sequence (s2n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (skn) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form (1/N)∑ Nn=1f1(Tsnx)f2(T2snx)⋯fℓ(Tℓsnx). We also prove a similar result for pointwise convergence of single ergodic averages.

scholarly journals Pointwise convergence of some multiple ergodic averages

2018 ◽  
Vol 330 ◽  
pp. 946-996 ◽  
Author(s):  
Sebastián Donoso ◽  
Wenbo Sun
Keyword(s):  
Pointwise Convergence ◽  
Ergodic Averages

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 Pointwise convergence of multiple ergodic averages and strictly ergodic models

2019 ◽  
Vol 139 (1) ◽  
pp. 265-305 ◽  
Author(s):  
Wen Huang ◽  
Song Shao ◽  
Xiangdong Ye
Keyword(s):  
Pointwise Convergence ◽  
Ergodic Averages

scholarly journals Non-commutative ergodic averages of balls and spheres over Euclidean spaces

2018 ◽  
Vol 40 (2) ◽  
pp. 418-436
Author(s):  
GUIXIANG HONG
Keyword(s):  
Ergodic Theorem ◽  
Pointwise Convergence ◽  
Dense Subset ◽  
Ergodic Theorems ◽  
Euclidean Spaces ◽  
Ergodic Averages ◽  
Transference Principle ◽  
Maximal Inequalities ◽  
Pointwise Ergodic Theorem ◽  
Special Case

In this paper, we establish a non-commutative analogue of Calderón’s transference principle, which allows us to deduce the non-commutative maximal ergodic inequalities from the special case—operator-valued maximal inequalities. As applications, we deduce the non-commutative Stein–Calderón maximal ergodic inequality and the dimension-free estimates of the non-commutative Wiener maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener’s pointwise ergodic theorem, following a somewhat standard way we construct a dense subset on which pointwise convergence holds. To show Jones’ pointwise ergodic theorem, we use again the transference principle together with the Littlewood–Paley method, which is different from Jones’ original variational method that is still unavailable in the non-commutative setting.

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