Related papers: Convergence of Diagonal Ergodic Averages

We study functional convergence of sums of moving averages with random coefficients and heavy-tailed innovations. Under some standard moment conditions and the assumption that all partial sums of the series of coefficients are a.s. bounded…

It is known that, for a positive Dunford-Schwartz operator in a noncommutative $L^p-$space, $1\leq p<\infty$ or, more generally, in a noncommutative Orlicz space with order continuous norm, the corresponding ergodic averages converge…

We present some twisted compactness conditions for almost everywhere convergence of one-parameter entangled ergodic averages of Dunford-Schwartz operators $T_0,\ldots, T_a$ on a Borel probability space of the form $$ \sum_{n=1}^N T_a^n…

By definition, a map quasiperiodic on a set $X$ if the map is conjugate to a pure rotation. Suppose we have a trajectory $(x_n)$ that we suspect is quasiperiodic. How do we determine if it is? In this paper we show how to compute the…

We prove a function-field analogue of Bourgain's $L^2$ pointwise ergodic theorem. Let $q$ be a power of a prime $p$, let $\mathbb{F}_q[t]$ be the ring of polynomials over the finite field $\mathbb{F}_q$, and let $\mathbb{F}_q[t][u]$ be the…

Motivated by Sarnak's conjecture on M\"obius orthogonality, we investigate the general problem of orthogonality for a bounded sequence to topological models of characteristic classes of measure-preserving automorphisms. Our main observation…

Let $(X,\mathcal{A}, \mu)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be invertible bi measurable measure preserving transformations on this measure space. We give a sufficient condition for the product of $H$ bounded…

In this article, we study the bilaterally almost uniform (b.a.u.) convergence of weighted averages of a positive Dunford-Schwartz operator on the noncommutative $L_p$-spaces associated to a semifinite von Neumann algebra by a large number…

Let $(X,T)$ be a topological dynamical system with metric $d$. We define a new function $\overline{F}(x,y)=\limsup\limits_{n \to +\infty} \inf\limits_{\sigma \in S_n} \frac 1n \sum\limits_{k=1}^n d(T^k x,T^{\sigma(k)} y)$ by using…

We study fluctuations of ergodic averages generated by actions of amenable groups. In the setting of an abstract ergodic theorem for locally compact second countable amenable groups acting on uniformly convex Banach spaces, we deduce a…

By employing an accelerated weighting method, we establish arbitrary polynomial and exponential pointwise convergence for multiple ergodic averages under general balancing conditions in both discrete and continuous settings, including…

In connection with the results of Tim Austin, and Wen Huang, Song Shao, Xiangdong Ye we present the following assertion: there are infinite automorphisms $S,T$, some set $A$ of positive finite measure and a sequence $N_m$ with…

We prove that the ergodic Ces\' aro averages generated by a positive Dunford-Schwartz operator in a noncommutative space $L^p(\mathcal M,\tau)$, $1<p<\infty$, converge almost uniformly (in Egorov's sense). This problem goes back to the…

Let $(X,\mu)$ be a probability space equipped with an invertible, measure-preserving transformation $T\colon X \to X$. We exhibit a wide class of weights $w$ so that whenever $f,g \in L^{\infty}(X)$, the bilinear ergodic averages \[…

Let $T$ be the map defined on $\N=\{1,2,3, ...\}$ by $T(n) = \frac{n}{2} $ if $n$ is even and by $T(n) = \frac{3n+1}{2}$ if $n$ is odd. Consider the dynamical system $(\N, 2^{\N}, T,\mu)$ where $\mu$ is the counting measure. This dynamical…

We show that every multi-correlation sequence is the sum of a generalized nilsequence and a null-sequence. This proves a conjecture of N. Frantzikinakis. A key ingredient is the reduction of ergodic multidimensional inverse theorems to…

Let $A$ be an expanding endomorphism on the torus ${\Bbb T}^d = {\Bbb R}^d /{\Bbb Z}^d$ with its smallest eigenvalue $\lambda >1$. Consider the ergodic system $({\Bbb T}^d, A, \mu)$ where $\mu$ is Haar measure. We prove that the correlation…

We introduce computable actions of computable groups and prove the following versions of effective Birkhoff's ergodic theorem. Let $\Gamma$ be a computable amenable group, then there always exists a canonically computable tempered two-sided…

Exploiting the recent work of Tao and Ziegler on a concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study criteria…

Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, and let $X\subset L^1(\Omega)+L^\infty(\Omega)$ be a fully symmetric space of measurable functions on $(\Omega,\mu)$. If $\mu(\Omega)=\infty$, necessary and sufficient conditions are…