Related papers: Koopman representations for positive definite func…

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…

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…

This paper follows on from our previous work, where we introduced the notion of \emph{confined extensions}, and our purpose is to widen the context in which such extensions appear. We do so in the setup of Poisson suspensions: we take a…

We show that, whenever Gamma is a countable abelian group and Delta is a finitely-generated subgroup of Gamma, a generic measure-preserving action of Delta on a standard atomless probability space (X,mu) extends to a free measure-preserving…

We investigate strongly separately continuous functions on a product of topological spaces and prove that if $X$ is a countable product of real lines, then there exists a strongly separately continuous function $f:X\to\mathbb R$ which is…

It is shown that there exist a probability space $(X,{\mathcal X},\mu)$, two ergodic measure preserving transformations $T,S$ acting on $(X,{\mathcal X},\mu)$ with $h_\mu(X,T)=h_\mu(X,S)=0$, and $f, g \in L^\infty(X,\mu)$ such that the…

Using percolation techniques, Gaboriau and Lyons recently proved that every countable, discrete, nonamenable group $\Gamma$ contains measurably the free group $\mathbf F_2$ on two generators: there exists a probability measure-preserving,…

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…

In [7], Kwapie\'{n} announced that every mean zero function $f\in L_\infty[0,1]$ can be written as a coboundary $f = g\circ T -g$ for some $g\in L_\infty[0,1]$ and some measure preserving transformation $T$ of $[0,1]$. Whereas the original…

We extend the Foia\c{s} and Stratila theorem to the case of $L^2$-functions whose spectral measure is continuous and concentrated on an independent Helson set, and to ergodic actions of locally compact second countable abelian groups. We…

Let $G$ and $H$ be locally compact, second countable groups. Assume that $G$ acts in a measure class preserving way on a standard probability space $(X,\mu)$ such that $L^\infty(X,\mu)$ has an invariant mean and that there is a Borel…

We prove a Khintchine-type recurrence theorem for pairs of endomorphisms of a countable discrete abelian group. As a special case of the main result, if $\Gamma$ is a countable discrete abelian group, $\varphi, \psi \in End(\Gamma)$, and…

We utilize Gaussian measure preserving systems to prove the existence and genericity of Lebesgue measure preserving transformations $T:[0,1]\rightarrow [0,1]$ which exhibit both mixing and rigidity behavior along families of asymptotically…

We prove that if \mu is a generically stable stable measure in a first order theory with NIP and mu(\phi(x,b)) = 0 for all b, then \mu^{(n)}(\exists y(\phi(x_1,y)\wedge ... \wedge \phi(x_n,y))) = 0. We deduce that if G is an fsg grooup then…

Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact sigma-compact…

Given a dynamical system $(X,f)$, we let $E(X,f)$ denote its Ellis semigroup and $E(X,f)^* = E(X,f) \setminus \{f^n : n \in \mathbb{N}\}$. We analyze the Ellis semigroup of a dynamical system having a compact metric countable space as a…

We define and study entanglement of continuous positive definite functions on products of compact groups. We formulate and prove an infinite-dimensional analog of Horodecki Theorem, giving a necessary and sufficient criterion for…

We give examples of rank-one transformations that are (weak) doubly ergodic and rigid (so all their cartesian products are conservative), but with non-ergodic $2$-fold cartesian product. We give conditions for rank-one infinite…

We show that $ { \omega }(n)$ and $ { \Omega }(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity are good weighting functions for the pointwise ergodic theorem…

A function $f:X\to Y$ between topological spaces is called {\em compact-preserving} if the image $f(K)$ of each compact subset $K\subset X$ is compact. We prove that a function $f:X\to Y$ defined on a strong Frechet space $X$ is…