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.

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 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.

The pointwise ergodic theorem in subsystems of second-order arithmetic

2007 ◽  
Vol 72 (1) ◽  
pp. 45-66 ◽  
Author(s):  
Ksenija Simic
Keyword(s):  
Ergodic Theorem ◽  
Second Order ◽  
Second Order Arithmetic ◽  
Pointwise Ergodic Theorem ◽  
Integrable Functions ◽  
Logical Strength ◽  
The Mean ◽  
Order Arithmetic ◽  
The One ◽  
Measure Preserving Transformations

AbstractThe pointwise ergodic theorem is nonconstructive. In this paper, we examine origins of this non-constructivity, and determine the logical strength of the theorem and of the auxiliary statements used to prove it. We discuss properties of integrable functions and of measure preserving transformations and give three proofs of the theorem, though mostly focusing on the one derived from the mean ergodic theorem. All the proofs can be carried out in ACA0; moreover, the pointwise ergodic theorem is equivalent to (ACA) over the base theory RCA0.

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

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.

scholarly journals Noncommutative weighted individual ergodic theorems with continuous time

2020 ◽  
Vol 23 (02) ◽  
pp. 2050013
Author(s):  
Vladimir Chilin ◽  
Semyon Litvinov
Keyword(s):  
Ergodic Theorem ◽  
Continuous Time ◽  
Symmetric Spaces ◽  
Von Neumann Algebra ◽  
Ergodic Theorems ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Von Neumann ◽  
Marcinkiewicz Spaces ◽  
Local Ergodic Theorem

We show that ergodic flows in the noncommutative [Formula: see text]-space (associated with a semifinite von Neumann algebra) generated by continuous semigroups of positive Dunford–Schwartz operators and modulated by bounded Besicovitch almost periodic functions converge almost uniformly. The corresponding local ergodic theorem is also proved. We then extend these results to arbitrary noncommutative fully symmetric spaces and present applications to noncommutative Orlicz (in particular, noncommutative [Formula: see text]-spaces), Lorentz, and Marcinkiewicz spaces. The commutative counterparts of the results are derived.

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