Related papers: Pointwise convergence of averages along cubes

Let $(X, \mathcal{B}, \mu)$ be a probability measure space and $T_1$, $T_2$, $T_3$ three not necessarily commuting measure preserving transformations on $(X, \mathcal{B}, \mu)$. We prove that for all bounded functions $f_1$, $f_2$, $f_3$…

Let $(X,\mathcal{B},\mu, T)$ be a measure preserving system. We prove the pointwise convergence of averages along cubes of $2^{k}-1$ bounded and measurable functions for all $k$.

We study the pointwise convergence of some weighted averages linked to averages along cubes. We show that if $(X,\mathcal{B},\mu, T_i)$ are not necessarily commuting measure preserving systems on the same finite measure space and if $f_i,$…

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 \[\frac{1}{N^3} \sum_{i,j,k=0}^{N-1} f_0(S^j T^k x) f_1 (S^{i+j} T^k x) f_2 (S^j T^{i+k} x)\]…

Let $(X,\mathcal{A}, \mu)$ be a probability measure space and let $T_i,$ $1\leq i\leq H,$ be commuting invertible measure preserving transformations on this measure space. We prove the following pointwise results; The averages…

We show that for every ergodic system $(X,\mu,T_1,\ldots,T_d)$ with commuting transformations, the average \[\frac{1}{N^{d+1}} \sum_{0\leq n_1,\ldots,n_d \leq N-1} \sum_{0\leq n\leq N-1} f_1(T_1^n \prod_{j=1}^d T_j^{n_j}x)f_2(T_2^n…

Huang, Shao and Ye recently studied pointwise multiple averages by using suitable topological models. Using a notion of dynamical cubes introduced by the authors, the Huang-Shao-Ye technique and the Host machinery of magic systems, we prove…

By building some suitable strictly ergodic models, we prove that for an ergodic system $(X,\mathcal{X},\mu, T)$, $d\in{\mathbb N}$, $f_1, \ldots, f_d \in L^{\infty}(\mu)$, the averages $$\frac{1}{N^2} \sum_{(n,m)\in [0,N-1]^2}…

We prove pointwise convergence, as $N\to \infty$, for the multiple ergodic averages $\frac{1}{N}\sum_{n=1}^N f(T^nx)\cdot g(S^{a_n}x)$, where $T$ and $S$ are commuting measure preserving transformations, and $a_n$ is a random version of the…

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…

We study pointwise convergence of entangled averages of the form \[ \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N} T_m^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\ldots A_2T_2^{n_{\alpha(2)}}A_1T_1^{n_{\alpha(1)}} f, \] where $f\in…

Fix $c\in (1,23/22)$. Let $\alpha$ and $\beta$ be two distinct non-zero real numbers with $|\alpha|\neq |\beta|$. It is shown that for any measure preserving system $(X,\mathcal{X},\mu,T)$ and any $f,g\in L^{\infty}(\mu)$, the limit…

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…

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 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…

Let $T_1, ..., T_l: X \to X$ be commuting measure-preserving transformations on a probability space $(X, \X, \mu)$. We show that the multiple ergodic averages $\frac{1}{N} \sum_{n=0}^{N-1} f_1(T_1^n x) ... f_l(T_l^n x)$ are convergent in…

Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-\alpha}$, $0 < \alpha < 1/2$. We prove that, almost surely, for every measure-preserving system…

For any measure preserving system $(X,\mathcal{B},\mu,T_1,\ldots,T_d),$ where we assume no commutativity on the transformations $T_i,$ $1\leq i\leq d,$ we study the pointwise convergence of multiple ergodic averages with iterates of…

We show that on a $\sigma$-finite measure preserving system $X = (X,\nu, T)$, the non-conventional ergodic averages $$ \mathbb{E}_{n \in [N]} \Lambda(n) f(T^n x) g(T^{P(n)} x)$$ converge pointwise almost everywhere for $f \in L^{p_1}(X)$,…

We establish a pointwise convergence result for ergodic averages modeled along orbits of the form $(n\lfloor n\sqrt{k}\rfloor)_{n\in\mathbb{N}}$, where $k$ is an arbitrary positive rational number with $\sqrt{k}\not\in\mathbb{Q}$. Namely,…