Related papers: Seminorm control for ergodic averages with commuti…

In this paper, for a discontinuous skew-product transformation with the integrable observation function, we obtain uniform ergodic theorem and semi-uniform ergodic theorem. The main assumptions are that discontinuity sets of transformation…

We prove the norm convergence of multiple ergodic averages along cubes for several commuting transformations, and derive corresponding combinatorial results. The method we use relies primarily on the "magic extension" established recently…

We provide necessary and sufficient conditions for joint ergodicity results for systems of commuting measure preserving transformations for an iterated Hardy field function of polynomial growth. Our method builds on and improves recent…

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 initiate the study of effective pointwise ergodic theorems in resource-bounded settings. Classically, the convergence of the ergodic averages for integrable functions can be arbitrarily slow. In contrast, we show that for a class of…

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 establish a number of "concatenation theorems" that assert, roughly speaking, that if a function exhibits "polynomial" (or "Gowers anti-uniform", "uniformly almost periodic", or "nilsequence") behaviour in two different directions…

Let $(X,\mathcal{B},\mu)$ be a probability space and let $T_1,..., T_l$ be $l$ commuting invertible measure preserving transformations \linebreak of $X$. We show that if $T_1^{c_1} ... T_l^{c_l}$ is ergodic for each $(c_1,...,c_l)\neq…

The joint ergodicity classification problem aims to characterize those sequences which are jointly ergodic along an arbitrary dynamical system if and only if they satisfy two natural, simpler-to-verify conditions on this system. These two…

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we prove that a bounded sequence is a good…

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…

In this paper, we extend the generalized Wiener-Wintner Theorem built by Host and Kra to the multilinear case under the hypothesis of pointwise convergence of multilinear ergodic averages. In particular, we have the following result: Let…

Let $T$ be a power bounded Hilbert space operator without unimodular eigenvalues. We show that the subsequential ergodic averages $N^{-1}\sum_{n=1}^N T^{a_n}$ converge in the strong operator topology for a wide range of sequences $(a_n)$,…

In the framework of statistical mechanics the properties of macroscopic systems are deduced starting from the laws of their microscopic dynamics. One of the key assumptions in this procedure is the ergodic property, namely the equivalence…

We apply Walsh's method for proving norm convergence of multiple ergodic averages to arbitrary amenable groups. We obtain convergence in the uniform Ces\`aro sense for their polynomial actions and for ``triangular'' averages associated to…

We investigate the limiting behavior of multiple ergodic averages along sparse sequences evaluated at prime numbers. Our sequences arise from smooth and well-behaved functions that have polynomial growth. Central to this topic is a…

Let $X$ be a one dimensional positive recurrent diffusion with initial distribution $\nu$ and invariant probability $\mu$. Suppose that for some $p> 1$, $\exists a\in\R$ such that $\forall x\in\R, \E_x T_a^p<\infty$ and $\E_\nu…

Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,\Sigma,\mu)$. We prove uniform extensions of the Wiener-Wintner theorem in two settings: For averages involving weights coming from Hardy field…

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.