Related papers: Convergence of Diagonal Ergodic Averages

Given sequence of measure preserving transformations $\{U_k:\,k=1,2,\ldots, n\}$ on a measurable space $(X,\mu)$. We prove a.e. convergence of the ergodic means \begin{equation} \frac{1}{s_1\cdots…

Consider a system $(X, \mathcal{F}, \mu, T)$, bounded functions $f_1, f_2 \in L^\infty(\mu)$ and $a,b \in \ZZ.$ We show that there exists a set of full measure $X_{f_1, f_2}$ in $X$ such that for all $x \in X_{f_1, f_2}$ and for every…

Let $S$ and $T$ be measure-preserving transformations of a probability space $(X,{\mathcal B},\mu)$. Let $f$ be a bounded measurable functions, and consider the integrals of the corresponding `double' ergodic averages:…

The purpose of this paper is to study ergodic averages with deterministic weights. More precisely we study the convergence of the ergodic averages of the type $\frac{1}{N} \sum_{k=0}^{N-1} \theta (k) f \circ T^{u_k}$ where $\theta = (\theta…

We prove a.e. convergence of continuous-time quadratic averages with respect to two commuting $\mathbb{R}$-actions, coming from a single jointly measurable measure-preserving $\mathbb{R}^2$-action on a probability space. The key ingredient…

We study the limiting behavior of multiple ergodic averages involving several not necessarily commuting measure preserving transformations. We work on two types of averages, one that uses iterates along combinatorial parallelepipeds, and…

We investigate the dynamics of continued fractions and explore the ergodic behaviour of the products of mixed partial quotients in continued fractions of real numbers. For any function $\Phi:\mathbb N\to [2,+\infty)$ and any integer $d\geq…

Let X=(X^0,\mu,T) be an ergodic measure preserving system. For a natural number k we consider the averages (*) 1/N \sum_{n=1}^N \prod_{j=1}^k f_j(T^{n a_j}x) where the functions f_j are bounded, and a_j are integers. A factor of X is…

We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges in $L^{2}$ to the…

We study the limiting behavior of multiple ergodic averages involving sequences of integers that satisfy some regularity conditions and have polynomial growth. We show that for "typical" choices of Hardy field functions $a(t)$ with…

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…

Let $F_2$ denote the free group on two generators $a,b$. For any measure-preserving system $(X, {\mathcal X}, \mu, (T_g)_{g \in F_2})$ on a finite measure space $X = (X,{\mathcal X},\mu)$, any $f \in L^1(X)$, and any $n \geq 1$, define the…

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…

For every $c\in(1,23/22)$ and every probability dynamical system $(X,\mathcal{B},\mu,T)$ we prove that for any $f,g\in L^{\infty}_{\mu}(X)$ the bilinear ergodic averages \[ \frac{1}{N}\sum_{n=1}^Nf(T^{\lfloor n^c\rfloor}x)g(T^{-\lfloor…

We establish a generalization of Anush Tserunyan and Jenna Zomback's 2024 Backward Ergodic Theorem. We remove the countable-to-one assumption and thus provide a backward ergodic theorem for arbitrary measure-preserving transformations.…

We show any subset $A\subset\mathbb{N}$ with positive upper Banach density contains the pattern $\{m,m+[n\alpha],\dots,m+k[n\alpha]\}$, for some $m\in\mathbb{N}$ and $n=p-1$ for some prime $p$, where…

This text is addressed to students. It is a short story about some problems in ergodic theory, both related and independent. We discuss the factorization of transformations into the product of three involutions; Furstenberg's theorem on…

In this paper, we investigate ergodic and fractal properties of the sets $$\Lambda_y:=\Big\{n\in\mathbb{N}:\ \{u_ny\}\in I_n\Big\},$$ where $\{\cdot\}$ denotes the fractional part function, $(u_n)_{n\in\mathbb{N}}$ is an increasing sequence…

We show that multiple polynomial ergodic averages arising from nilpotent groups of measure preserving transformations of a probability space always converge in the L^2 norm.

We prove that for every $c\in(1,23/22)$, every probability space $(X,\mathcal{B},\mu)$ equipped with two commuting measure-preserving transformations $T,S\colon X\to X$ and every $f,g\in L^{\infty}_{\mu}(X)$ we have that the…