Related papers: Koopman representations for positive definite func…

A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\mathcal{B},\mu,T)$ and any bounded measurable function $f$, the averages $ \frac1N \sum_{n=1}^N f(T^{s_n}x)$ converge…

Let $\mathcal{M}(X,\mathcal{A},\mu)$ be the ring of all real-valued measurable functions constructed over a measure space $(X,\mathcal{A},\mu)$. A topology on $\mathcal{M}(X,\mathcal{A},\mu)$, called the {$F_\mu$-topology} weaker than the {…

Let $\mu$ be the Haar measure of a unimodular locally compact group $G$ and $m (G)$ as the infimum of the volumes of all open subgroups of $G$. The main result of this paper is that \begin{align*} \int_{G}^{} f \circ \left( \phi_1 * \phi_2…

Building on previous work in the nilspace-theoretic approach to the study of Host-Kra factors of measure-preserving systems, we prove that every ergodic $\mathbb{F}_p^\omega$-system of order $k$ is a factor of an Abramov…

We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation $(X, E)$ may be realized as the topological ergodic decomposition of a continuous action of a countable group $\Gamma…

Let $(M,g)$ be a two dimensional compact Riemannian manifold of genus $g(M)>1$. Let $f$ be a smooth function on $M$ such that $$f \ge 0, \quad f\not\equiv 0, \quad \min_M f = 0. $$ Let $p_1,\ldots,p_n$ be any set of points at which…

Our aim is to find the irreducibility criteria for the Koopman representation, when the group acts on some space with a measure (Conjecture 1.5). Some general necessary conditions of the irreducibility of this representation are…

We consider infinite measure-preserving non-primitive self-similar tiling systems in Euclidean space $\mathbb R^d$. We establish the second-order ergodic theorem for such systems, with exponent equal to the Hausdorff dimension of a…

We prove that a Kleinian group $G$ acting upon $\mathbb{H}^{n}$ admits a non-constant $G$-automorphic function, even if it has torsion elements, provided that the orders of the elliptic (torsion) elements are uniformly bounded. This is…

We consider volume-preserving flows $(\Phi^f_t)_{t\in\mathbb{R}}$ on $S\times \mathbb{R}$, where $S$ is a closed connected surface of genus $g\geq 2$ and $(\Phi^f_t)_{t\in\mathbb{R}}$ has the form $\Phi^f_t(x,y)=(\phi_tx,y+\int_0^t…

It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system the Birkhoff average of every integrable function is almost…

We find sufficient conditions on a compactly supported function $g$, $\supp g = [a,b]$ which guarantee that the Gabor system $$\mathcal{G}(g;\alpha,\beta)=\{e^{2\pi i \beta m x}g(x-\alpha n)\}_{m,n\in\mathbb{Z}}$$ is a frame for all $\alpha…

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}…

Bochner's theorem says that if $M$ is a compact Riemannian manifold with negative Ricci curvature, then the isometry group $\operatorname{Iso}(M)$ is finite. In this article, we show that if $(X,d,m)$ is a compact metric measure space with…

We study two properties of nonsingular and infinite measure-preserving ergodic systems: weak double ergodicity, and ergodicity with isometric coefficients. We show that there exist infinite measure-preserving transformations that are…

Let $G$ be a connected compact group equipped with the normalised Haar measure $\mu$. Our first result shows that given $\alpha, \beta>0$, there is a constant $c = c(\alpha,\beta)>0$ such that for any compact sets $A,B\subseteq G$ with $…

The Mackey-Zimmer theorem classifies ergodic group extensions $X$ of a measure-preserving system $Y$ by a compact group $K$, by showing that such extensions are isomorphic to a group skew-product $X \equiv Y \rtimes_\rho H$ for some closed…

Given an ergodic probability measure preserving dynamical system $\G\acts (X,\mu)$, where $\G$ is a finitely generated countable group, we show that the asymptotic growth of the number of finite models for the dynamics, in the sense of…

We proved that for the countably infinite number of one-parameterized one dimensional dynamical systems, they preserve the Lebesgue measure and they are ergodic for the measure (infinite ergodicity). Considered systems connect the parameter…

Let $f$ be a $C^{1+\alpha}$ diffeomorphism of a compact Riemannian manifold and $\mu$ an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential $\phi$ there exists a sequence of basic sets $\Omega_n$…