Related papers: Individual ergodic theorems for infinite measure

Given a probability space $(X,\mu)$, a square integrable function $f$ on such space and a (unilateral or bilateral) shift operator $T$, we prove under suitable assumptions that the ergodic means $N^{-1}\sum_{n=0}^{N-1} T^nf$ converge…

Using tools from the theory of optimal transport, we establish several results concerning isometric actions of amenable topological groups with potentially unbounded orbits. Specifically, suppose $d$ is a compatible left-invariant metric on…

We show that for any C^1+alpha diffeomorphism of a compact Riemannian manifold, every non-atomic, ergodic, invariant probability measure with non-zero Lyapunov exponents is approximated by uniformly hyperbolic sets in the sense that there…

We show that for every ergodic system $(X,\mu,T_1,\ldots,T_d)$ with commuting transformations, the average \[\frac{1}{N^{d+1}} \sum_{0\leq n_1,\ldots,n_d \leq N-1} \sum_{0\leq n\leq N-1} f_1(T_1^n \prod_{j=1}^d T_j^{n_j}x)f_2(T_2^n…

Let $U$ be a unitary operator acting on the Hilbert space $\ch$, and $\a:\{1,..., 2k\}\mapsto\{1,..., k\}$ a pair partition. Then the ergodic average $$ \frac{1}{N^{k}}\sum_{n_{1},...,n_{k}=0}^{N-1} U^{n_{\a(1)}}A_{1}U^{n_{\a(2)}}...…

Suppose that $\Omega \subset\mathbb R^{n+1}$, $n\geq1$, is a uniform domain with $n$-Ahlfors regular boundary and $L$ is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in $\Omega$. We show that the…

Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $\Omega: \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+\delta_\Omega$, where $\delta_\Omega>0$ is small. Assume also that $\Omega$ is twice…

We study a one-parameter family of interval maps $\{T_\alpha\}_{\alpha\in[1,\beta]}$, with $\beta$ the golden mean, defined on $[-1,1]$ by $T_\alpha(x)=\beta^{1+|t|}x-t\beta\alpha$ where $t\in\{-1,0,1\}$. For each $T_\alpha,\ \alpha>1$, we…

This work aims to investigate the existence of ergodic invariant measures and its uniqueness, associated with obstacle problems governed by a T-monotone operator defined on Sobolev spaces and driven by a multiplicative noise in a bounded…

Based on T.Tao's result of norm convergence of multiple ergodic averages for commut-ing transformation, we obtain there is a subsequence which converges almost everywhere. Meanwhile, the ergodic behaviour, which the time average is equal to…

Let $(X,\mu)$ be an arbitrary measure space equipped with a family of pairwise commuting measure preserving transformations $T_1, \dotsc, T_m$. We prove that the ergodic averages \[ A_{N;X}^{P_1, \dotsc, P_m}f = \frac{1}{N} \sum_{n=1}^N…

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…

Let (G,+) be a compact, abelian, and metrizable topological group. In this group we take $g\in G$ such that the corresponding automorphism t_g is ergodic. The main result of this paper is a new ergodic theorem for functions in L^1(G,M),…

The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is…

Urbanik's theorem for a Poisson process on an infinite measure space (X, A, $\mu$) relates integrability of stochastic integrals to a particular Orlicz function space L$\Phi$ ($\mu$) on which the L1-norm of the Poisson process induces a…

In a domain $\Omega\subset \mathbb{R}^{\mathbf{N}}$ we consider a selfadjoint operator $\mathbf{T}=\mathfrak{A}^*P\mathfrak{A} ,$ where $\mathfrak{A}$ is a pseudodifferential operator of order $-l=-\mathbf{N}/2$ and $P=V\mu_{\Sigma}$ is a…

We extend the so-called Kotani Theory for a particular class of ergodic matrix-like Jacobi operators defined in $l^{2}(\mathbb{Z}; \mathbb{C}^{l})$ by the law $[H_{\omega} \textbf{u}]_{n} := D^{*}(T^{n - 1}\omega) \textbf{u}_{n - 1} +…

We prove that for every $c\in(1,23/22)$, every probability space $(X,\mathcal{B},\mu)$ equipped with two commuting measure-preserving transformations $T,S\colon X\to X$ and every $f,g\in L^{\infty}_{\mu}(X)$ we have that the…

In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace-Beltrami operator on the unit sphere. The sufficient conditions for summability is obtained. The more general…

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 $f \in L^{p_1}(X)$,…