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.

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 $L^{p}$, $p>1$.

Weighted and Subsequential Ergodic Theorems

1983 ◽  
Vol 35 (1) ◽  
pp. 145-166 ◽  
Author(s):  
J. R. Baxter ◽  
J. H. Olsen
Keyword(s):  
Linear Operator ◽  
Ergodic Theorem ◽  
Probability Space ◽  
Complex Numbers ◽  
Ergodic Theorems ◽  
Pointwise Ergodic Theorem ◽  
Image Position

1. Introduction. Let (X, , μ) be a probability space, T a linear operator on ℒp(X, , μ), for some p, 1 ≦ p ≦ ∞. Let an be a sequence of complex numbers, n = 0, 1, …, which we shall often refer to as weights. We shall say that the weighted pointwise ergodic theorem holds for T on ℒp, if, for every ƒ in ℒp,1.1Let a denote the sequence (an). If (1.1) holds we shall say that a is Birkhoff for T on ℒp, or, more briefly, that (a, T) is Birkhoff.We are also interested in ergodic theorems for subsequences. Let n(k) be a subsequence. We shall say the pointwise ergodic theorem holds for the subsequence n(k) and the operator T if, for every ƒ in ℒp,1.2

Multiparameter Weighted Ergodic Theorems

1994 ◽  
Vol 46 (2) ◽  
pp. 343-356 ◽  
Author(s):  
Roger L. Jones ◽  
James Olsen
Keyword(s):  
Ergodic Theorem ◽  
Point Of View ◽  
Ergodic Theorems ◽  
Natural Class ◽  
Pointwise Ergodic Theorem ◽  
Point Transformations

AbstractIn this paper we show that multi-dimensional bounded Besicovitch weights are good weights for the pointwise ergodic theorem for Dunford-Schwartz operators and positively dominated contractions of LP. This in particular implies new weighted results for multi-parameter measure preserving point transformations. The proofs show that Besicovitch weights are a very natural class when considered from the operator point of view. We also show that for 1 ≤ r < ∞, the r-bounded Besicovitch classes are all the same, generalizing a result of Bellow and Losert.

scholarly journals Quantitative ergodic theorems for weakly integrable functions

2012 ◽  
Vol 34 (2) ◽  
pp. 534-542 ◽  
Author(s):  
ALAN HAYNES
Keyword(s):  
Ergodic Theorem ◽  
Ergodic Theorems ◽  
Pointwise Ergodic Theorem ◽  
Integrable Functions

AbstractUnder suitable hypotheses we establish a quantitative pointwise ergodic theorem which applies to trimmed Birkhoff sums of weakly integrable functions.

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.

Oscillation in ergodic theory

1998 ◽  
Vol 18 (4) ◽  
pp. 889-935 ◽  
Author(s):  
ROGER L. JONES ◽  
ROBERT KAUFMAN ◽  
JOSEPH M. ROSENBLATT ◽  
MÁTÉ WIERDL
Keyword(s):  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Square Function ◽  
Ergodic Averages ◽  
Pointwise Ergodic Theorem ◽  
Additional Information

In this paper we establish a variety of square function inequalities and study other operators which measure the oscillation of a sequence of ergodic averages. These results imply the pointwise ergodic theorem and give additional information such as control of the number of upcrossings of the ergodic averages. Related results for differentiation and for the connection between differentiation operators and the dyadic martingale are also established.

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

2021 ◽  
pp. 1-36
Author(s):  
ARIE LEVIT ◽  
ALEXANDER LUBOTZKY
Keyword(s):  
Ergodic Theorem ◽  
Abelian Groups ◽  
Wreath Products ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Amenable Groups ◽  
Pointwise Ergodic Theorem ◽  
Lamplighter Group ◽  
Invariant Random Subgroups

Abstract We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

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 A pointwise ergodic theorem for Fuchsian groups

2011 ◽  
Vol 151 (1) ◽  
pp. 145-159 ◽  
Author(s):  
ALEXANDER I. BUFETOV ◽  
CAROLINE SERIES
Keyword(s):  
Ergodic Theorem ◽  
Fuchsian Group ◽  
Fuchsian Groups ◽  
Limit Sets ◽  
Pointwise Ergodic Theorem ◽  
Cesaro Averages

AbstractWe use Series' Markovian coding for words in Fuchsian groups and the Bowen-Series coding of limit sets to prove an ergodic theorem for Cesàro averages of spherical averages in a Fuchsian group.

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