Related papers: Seminorm control for ergodic averages with commuti…

We establish mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive…

A pseudorandom point in an ergodic dynamical system over a computable metric space is a point which is computable but its dynamics has the same statistical behavior as a typical point of the system. It was proved in [Avigad et al. 2010,…

We offer a generalization of the recent result of Tao (building on earlier results of Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler) that the nonconventional ergodic averages associated to…

We establish pointwise convergence for nonconventional ergodic averages taken along $\lfloor p^c\rfloor$, where $p$ is a prime number and $c\in(1,4/3)$ on $L^r$, $r\in(1,\infty)$. In fact, we consider averages along more general sequences…

We establish pointwise almost everywhere convergence for the polynomial multiple ergodic averages $$\frac{1}{N} \sum_{n=1}^N \La(n) f_1(T^{P_1(n)} x)\cdots f_k(T^{P_k(n)} x)$$ as $N\to \infty$, where $\La$ is the von Mangoldt function, $T…

For a Dunford-Schwartz operator in a fully symmetric space of measurable functions of an arbitrary measure space, we prove pointwise convergence of the conventional and weighted ergodic averages.

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an…

For a continuous semicascade on a metrizable compact set $\Omega $, we consider the weak$^{*}$ convergence of generalized operator ergodic means in ${\rm End}\, \, C^{*} (\Omega)$. We discuss conditions on the dynamical system under which…

A joint measure-preserving system is $(X, \mathcal{B}, \mu_{1}, \dots, \mu_{k}, T_{1}, \dots, T_{k})$, where each $(X, \mathcal{B}, \mu_{i}, T_{i})$ is a measure-preserving system and any $\mu_{i}$ and $\mu_{j}$ are mutually absolutely…

Consensus is a well-studied problem in distributed sensing, computation and control, yet deriving useful and easily computable bounds on the rate of convergence to consensus remains a challenge. This paper discusses the use of seminorms for…

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…

This paper investigates the ergodicity of Markov--Feller semigroups on Polish spaces, focusing on very weak regularity conditions, particularly the Ces\`aro eventual continuity. First, it is showed that the Ces\`aro average of such…

The idea of a parsing of a stationary process according to a collection of words is introduced, and the basic framework required for the asymptotic analysis of these parsings is presented. We demonstrate how the pointwise ergodic theorem…

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…

We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure preserving $\mathbb{Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and a null…

For $i = 0, 1, 2, \dots, k$, let $\mu_i$ be a Borel probability measure on $[0,1]$ which is equivalent to Lebesgue measure $\lambda$ and let $T_i:[0,1] \rightarrow [0,1]$ be $\mu_i$-preserving ergodic transformations. We say that…

The \emph{Gowers uniformity norms} $\|f\|_{U^k(G)}$ of a function $f: G \to \C$ on a finite additive group $G$, together with the slight variant $\|f\|_{U^k([N])}$ defined for functions on a discrete interval $[N] := \{1,...,N\}$, are of…

The Birkhoff Ergodic Theorem concludes that time averages, that is, Birkhoff averages, $\Sigma_{n=1}^N f(x_n)/N$ of a function $f$ along an ergodic trajectory $(x_n)$ of a function $T$ converges to the space average $\int f d\mu$, where…

Let $(X,\mu)$ be an arbitrary measure space equipped with a family of pairwise commuting measure preserving transformations $T_1, \dotsc, T_m$. We prove that the ergodic averages \[ A_{N;X}^{P_1, \dotsc, P_m}f = \frac{1}{N} \sum_{n=1}^N…

We show that if $(X,\mathcal{X},\mu,S,T)$ is an ergodic measure preserving system with commuting transformations $S$ and $T$, then the average \[\frac{1}{N^3} \sum_{i,j,k=0}^{N-1} f_0(S^j T^k x) f_1 (S^{i+j} T^k x) f_2 (S^j T^{i+k} x)\]…