Related papers: Ergodic Theorems for PSPACE functions and their co…

For a jointly measurable probability-preserving action $\tau:\mathbb{R}^D\curvearrowright (X,\mu)$ and a tuple of polynomial maps $p_i:\mathbb{R}\to \mathbb{R}^D$, $i=1,2,...,k$, the multiple ergodic averages \[ \frac{1}{T}\int_0^T…

Motivated by the well-known phase-space portrait of the nonlinear pendulum, the purpose of this paper is to obtain convergence rates in the ergodic theorem for flows in the plane that have arbitrarily slow trajectories. Considering bounded…

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…

We prove essentially optimal $L^p(\mathbb{R})$-estimates for variational variants of the maximal Fourier multiplier operators considered by Bourgain in his work on pointwise convergence of polynomial ergodic averages. As a corollary of our…

Let $(X, \mathcal{A},\mu)$ be a probability space and let $T$ be a contraction on $L^2(\mu)$. We provide suitable conditions over sequences $(w_k)$, $(u_k)$ and $(A_k)$ in such a way that the weighted ergodic limit…

Given an ergodic dynamical system $(X, \mathcal{B}, \mu, T)$, we prove that for each function $f$ belonging to the Orlicz space $L(\log L)^2(\log \log L)(X, \mu)$, the ergodic averages \[ \frac{1}{\pi(N)} \sum_{p \in \mathbb{P}_N} f\big(T^p…

We show the failure of the pointwise convergence of averages along the Omega function in a number field. As a consequence, we show, for instance, that the averages \[ \frac{1}{N^2}\sum_{1\leq m,n \leq N} f(T^{\Omega(m^2+n^2)}x)\] do not…

We prove a new pointwise ergodic theorem for probability-measure-preserving (pmp) actions of free groups, where the ergodic averages are taken over arbitrary finite subtrees of the standard Cayley graph rooted at the identity. This result…

Hopf's ratio ergodic theorem has an inherent symmetry which we exploit to provide a simplification of standard proofs of Hopf's and Birkhoff's ergodic theorems. We also present a ratio ergodic theorem for conservative transformations on a…

This paper resolves the question of pointwise convergence for ergodic averages of a single function along the set of polynomial values of primes of the form $x^2 + ny^2$. Following the influential paper of Bourgain…

We provide a unified framework to proving pointwise convergence of sparse sequences, deterministic and random, at the $L^1(X)$ endpoint. Specifically, suppose that \[ a_n \in \{ \lfloor n^c \rfloor, \min\{ k : \sum_{j \leq k} X_j = n\} \}…

We study in this paper the validity of the mean ergodic theorem along \emph{left} F\o lner sequences in a countable amenable group $G$. Although the \emph{weak} ergodic theorem always holds along \emph{any} left F\o lner sequence in $G$, we…

We study the convergence of the so-called entangled ergodic averages $\frac{1}{N^k}\sum_{n_1,...,n_k=1}^{N}T_m^{n_{\alpha(m)}}A_{m-1}T_{m-1}^{n_{\alpha(m-1)}}A_{m-2}...A_1T_1^{n_{\alpha(1)}},$ where $k\leq m$ and…

In this paper, we establish the quantitative mean ergodic theorems for two subclasses of power bounded operators on a fixed noncommutative $L_p$-space with $1<p<\infty$, which mainly concerns power bounded invertible operators and Lamperti…

Amos Nevo established the pointwise ergodic theorem in $L^p$ for measure-preserving actions of $\mathrm{PSL}_2(\mathbb{R})$ on probability spaces with respect to ball averages and every $p>1$. This paper shows by explicit example that…

Let $L^2(X,\Sigma,\mu,\tau)$ be a measure-preserving system, with $\tau$ a $\mathbb{Z}$-action. In this note, we prove that the ergodic averages along integer-valued polynomials, $P(n)$, \[ M_N(f):= \frac{1}{N}\sum_{n \leq N} \tau^{P(n)} f…

We generalize stochastic subgradient descent methods to situations in which we do not receive independent samples from the distribution over which we optimize, but instead receive samples that are coupled over time. We show that as long as…

We investigate pointwise convergence of entangled ergodic averages of Dunford-Schwartz operators $T_0,T_1,\ldots, T_m$ on a Borel probability space. These averages take the form \[ \frac{1}{N^k}\sum_{1\leq n_1,\ldots, n_k\leq N}…

We examine the convergence of ergodic averages along polynomials in Toeplitz systems and prove that it is possible for averages along one polynomial to converge, and along another to diverge. We also study density of the polynomial orbits…

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.