scholarly journals Effective results on nonlinear ergodic averages in CAT spaces

2015 ◽  
Vol 36 (8) ◽  
pp. 2580-2601 ◽  
Author(s):  
LAURENŢIU LEUŞTEAN ◽  
ADRIANA NICOLAE
Keyword(s):  
Ergodic Theorem ◽  
Asymptotic Regularity ◽  
Ergodic Averages ◽  
Von Neumann ◽  
Quantitative Version ◽  
Halpern Iteration ◽  
Mean Ergodic Theorem ◽  
Proof Mining ◽  
Nonlinear Generalization ◽  
Highly Uniform

In this paper we apply proof mining techniques to compute, in the setting of CAT$(\unicode[STIX]{x1D705})$ spaces (with $\unicode[STIX]{x1D705}>0$), effective and highly uniform rates of asymptotic regularity and metastability for a nonlinear generalization of the ergodic averages, known as the Halpern iteration. In this way, we obtain a uniform quantitative version of a nonlinear extension of the classical von Neumann mean ergodic theorem.

scholarly journals A quantitative mean ergodic theorem for uniformly convex Banach spaces

2009 ◽  
Vol 29 (6) ◽  
pp. 1907-1915 ◽  
Author(s):  
U. KOHLENBACH ◽  
L. LEUŞTEAN
Keyword(s):  
Banach Spaces ◽  
Ergodic Theorem ◽  
Hilbert Spaces ◽  
Local Stability ◽  
Uniformly Convex ◽  
Norm Convergence ◽  
Ergodic Averages ◽  
Mean Ergodic Theorem ◽  
Uniformly Convex Banach Spaces ◽  
The Mean

AbstractWe provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of Tao of the mean ergodic theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by Avigad et al [Local stability of ergodic averages. Trans. Amer. Math. Soc. to appear] and Tao [Norm convergence of multiple ergodic averages for commuting transformations. Ergod. Th. & Dynam. Sys.28(2) (2008), 657–688].

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.

A mean ergodic theorem of an amenable group action

2014 ◽  
Vol 17 (01) ◽  
pp. 1450003 ◽  
Author(s):  
Anilesh Mohari
Keyword(s):  
Ergodic Theorem ◽  
Group Action ◽  
Von Neumann Algebra ◽  
Amenable Group ◽  
Ergodic Theorems ◽  
Von Neumann ◽  
Mean Ergodic Theorem ◽  
Faithful Normal State ◽  
Contractive Maps ◽  
Dual Banach Space

We consider a sequence of weak Kadison–Schwarz maps τn on a von-Neumann algebra ℳ with a faithful normal state ϕ sub-invariant for each (τn, n ≥ 1) and use a duality argument to prove strong convergence of their pre-dual maps when their induced contractive maps (Tn, n ≥ 1) on the GNS space of (ℳ, ϕ) are strongly convergent. The result is applied to deduce improvements of some known ergodic theorems and Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space ℳ*.

scholarly journals A MEAN ERGODIC THEOREM FOR ACTIONS OF AMENABLE QUANTUM GROUPS

2008 ◽  
Vol 78 (1) ◽  
pp. 87-95 ◽  
Author(s):  
ROCCO DUVENHAGE
Keyword(s):  
Ergodic Theorem ◽  
Quantum Groups ◽  
Von Neumann Algebra ◽  
Weak Form ◽  
Locally Compact ◽  
Von Neumann ◽  
Locally Compact Quantum Groups ◽  
Mean Ergodic Theorem ◽  
Compact Quantum Groups ◽  
The Mean

AbstractWe prove a weak form of the mean ergodic theorem for actions of amenable locally compact quantum groups in the von Neumann algebra setting.

Mean ergodic theorem in function symmetric spaces for infinite measure

2018 ◽  
Vol 2018 (1) ◽  
pp. 35-46
Author(s):  
Vladimir Chilin ◽  
◽  
Aleksandr Veksler ◽  
Keyword(s):  
Ergodic Theorem ◽  
Symmetric Spaces ◽  
Infinite Measure ◽  
Mean Ergodic Theorem

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

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