Related papers: Pointwise convergence of some multiple ergodic ave…

We prove a pointwise ergodic theorem and a maximal inequality for actions of amenable groups on noncommutative measure spaces. To do so, we establish a square function estimate quantifying the difference between ergodic averages and some…

We prove that the divisor function $d(n)$ counting the number of divisors of the integer $n$, is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system $(X, {\mathcal A},\nu,\tau)$ and any $f\in…

We study multiple ergodic averages along IP sets, meaning we restrict iterates in the averages to all finite sums of some infinite sequence of natural numbers. We give criteria for convergence and divergence in mean of these multiple…

Examining multiple ergodic averages whose iterates are integer parts of real valued polynomials for totally ergodic systems, we provide various characterizations of total joint ergodicity, meaning that an average converges to the "expected"…

In this survey we present some recent applications of proof mining to the fixed point theory of (asymptotically) nonexpansive mappings and to the metastability (in the sense of Terence Tao) of ergodic averages in uniformly convex Banach…

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…

These notes are based on a course for a general audience given at the Centro de Modeliamento Matem\'atico of the University of Chile, in December 2004. We study the mean convergence of multiple ergodic averages, that is, averages of a…

We use techniques of proof mining to obtain a computable and uniform rate of metastability (in the sense of Tao) for the mean ergodic theorem for a finite number of commuting linear contractive operators on a uniformly convex Banach space.

Let $(X,\Sigma,m,\tau)$ be an ergodic system, that is, $(X, \Sigma, m)$ is a probability space and $\tau: X \to X$ is an invertible ergodic $m$-preserving transformation. For a function $f:X\to\mathbb R$, let $A_Nf$ denote the $N$th ergodic…

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…

We find limits of some multiple ergodic averages, generalizing a result of Bergelson to the setting of two commuting transformations and actions of amenable groups.

A celebrated result by Bourgain and Wierdl states that ergodic averages along primes converge almost everywhere for $L^p$-functions, $p>1$, with a polynomial version by Wierdl and Nair. Using an anti-correlation result for the von Mangoldt…

The main goal of the paper is to prove convergence in norm and pointwise almost everywhere on $L^p$, $p\in (1,\infty)$, for certain multiparameter polynomial ergodic averages in the spirit of Dunford and Zygmund for continuous flows. We…

We prove a multidimensional ergodic theorem with weighted averages for the action of the group $\mathbb{Z}^d$ on a probability space. At level $n$ weights are of the form $n^{-d} \psi(j/n)$, $ j\in \mathbb{Z}^d$, for real functions $\psi$…

We study the almost sure convergence of bilateral ergodic averages for not necessarily integrable functions and relate it to the ones of the forward and backward averages, hence complementing results of Wo\'s and the second named author. In…

We study random exponential sums of the form $\sum_{k=1}^nX_k\times\ex p\{i(\lambda_k^{(1)}t_1+...+\lambda_k^{(s)}t_s)\}$, where $\{X_n\}$ is a sequence of random variables and $\{\lambda_n^{(i)}:1\leq i\leq s\}$ are sequences of real…

Recently, Kiriki, Nakano and Soma introduced a concept called pointwise emergence as a new quantitative perspective into the study of non-existence of averages for dynamical systems. In the present paper, we consider the set of points with…

Consider a class of skew product transformations consisting of an ergodic or a periodic transformation on a probability space (M, B, m) in the base and a semigroup of transformations on another probability space (W,F,P) in the fibre. Under…

We develop a framework for the study of the limiting behavior of multiple ergodic averages with commuting transformations when all iterates are given by the same sparse sequence; this enables us to partially resolve several longstanding…

Suppose that G is a compact Abelian topological group, m is the Haar measure on G and f is a measurable function. Given (n_k), a strictly monotone increasing sequence of integers we consider the nonconventional ergodic/Birkhoff averages…