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A celebrated result of Hal\'asz describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions…

The goal is to show that an edge-reinforced random walk on a graph of bounded degree, with reinforcement weight function $W$ taken from a general class of reciprocally summable reinforcement weight functions, traverses a random attracting…

We present a general fixed point theorem which can be seen as the quintessence of the principles of proof for Banach's Fixed Point Theorem, ultrametric and certain topological fixed point theorems. It works in a minimal setting, not…

This article studies a structural aspect of measure-preserving actions of products of countable discrete groups, involving a so-called 'synergodic decomposition' in terms of the ergodic components of the actions of the two factor groups. We…

We prove a ratio ergodic theorem for non-singular free $Z^d$ and $R^d$ actions, along balls in an arbitrary norm. Using a Chacon-Ornstein type lemma the proof is reduced to a statement about the amount of mass of a probability measure that…

We consider a point process sequence induced by a stationary symmetric alpha-stable (0 < alpha < 2) discrete parameter random field. It is easy to prove, following the arguments in the one-dimensional case in Resnick and Samorodnitsky…

The stability against perturbations of a dynamical system conserving a generalized phase-space volume is studied by exploiting the similarity between statistical physics formalism and that of ergodic theory. A general continuity theorem is…

Exploiting the recent work of Tao and Ziegler on a concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study criteria…

We study multiple ergodic averages along IP sets, meaning we restrict iterates in the averages to all finite sums of some infinite sequence of natural numbers. We give criteria for convergence and divergence in mean of these multiple…

Let $(X,\mu)$ be a probability space, $G$ a countable amenable group and $(F_n)_n$ a left F\o lner sequence in $G$. This paper analyzes the non-conventional ergodic averages \[\frac{1}{|F_n|}\sum_{g \in F_n}\prod_{i=1}^d (f_i\circ…

In this paper, we prove a new ergodic theorem for $\mathbb{R}^d$-actions involving averages over dilated submanifolds, thereby generalizing the theory of spherical averages. Our main result is a quantitative estimate for the error term of…

We prove the uniform $\ell^2$-valued maximal inequalities for polynomial ergodic averages and truncated singular operators of Cotlar type modeled over multi-dimensional subsets of primes. In the averages case, we combine this with earlier…

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 will prove an S-arithmetic version of a theorem of Dani-Margulis on the convergence of ergodic averages of a given bounded continuous function, when the initial point is outside certain compact subsets of the singular set associated to…

We prove non-convergence theorems towards an unstable equilibrium (or a trap) for stochastic processes. The processes we consider are continuous-time or discrete-time processes and can be pertubations of the flow generated by a vector…

We study an intermittent quasistatic dynamical system composed of nonuniformly hyperbolic Pomeau--Manneville maps with time-dependent parameters. We prove an ergodic theorem which shows almost sure convergence of time averages in a certain…

The paper is primarily concerned with the asymptotic behavior as $N\to\infty$ of averages of nonconventional arrays having the form $N^{-1}\sum_{n=1}^N\prod_{j=1}^\ell T^{P_j(n,N)}f_j$ where $f_j$'s are bounded measurable functions, $T$ is…

We obtain the following embedding theorem for symbolic dynamical systems. Let $G$ be a countable amenable group with the comparison property. Let $X$ be a strongly aperiodic subshift over $G$. Let $Y$ be a strongly irreducible shift of…

In this paper, we reduce pointwise convergence of polynomial ergodic averages of general measure-preserving system acted by $\mathbb{Z}^{d}$ to the case of measure-preserving system acted by $\mathbb{Z}^{d}$ with zero entropy. As an…

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,…