Related papers: Diffuse Behaviour of Ergodic Sums Over Rotations

Let $x \mapsto x+ \alpha$ be a rotation on the circle and let $\varphi$ be a step function. We denote by $\varphi\_n (x)$ the corresponding ergodic sums $\sum\_{j=0}^{n-1} \varphi(x+j \alpha)$. Under an assumption on $\alpha$, for example…

For $R_{\alpha}$ being an irrational rotation of angle $\alpha$ on the one torus $\mathbb{T}$ and $\phi(x)=\frac{1}{x}-\frac{1}{1-x}$, we compare the behavior of the Birkhoff sum $S_N(\phi)=\sum_{k=0}^{N-1}(\phi\circ R_{\alpha}^k)(x)$ with…

We show that there are an irrational rotation $Tx=x+\alpha$ on the circle $\mathbb{T}$ and a continuous $\varphi\colon\mathbb{T}\to\mathbb{R}$ such that for each (continuous) uniquely ergodic flow $\mathcal{S}=(S_t)_{t\in\mathbb{R}}$ acting…

We consider classical dynamical properties of a particle in a constant gravitational force and making specular reflections with circular, elliptic or oval boundaries. The model and collision map are described and a detailed study of the…

We prove some results on the behavior of infinite sums of the form $\Sigma f\circ T^n(x)\frac{1}{n}$, where $T:S^1\to S^1$ is an irrational circle rotation and $f$ is a mean-zero function on $S^1$. In particular, we show that for a certain…

We establish functional limit theorems for ergodic sums of observables with power singularities for expanding circle maps. In the regime where the observables have infinite variance, we show that when rescaled by $N^{1/s}(\ln N)^\alpha$,…

Let T_\alpha denote the rotation T_{\alpha}x=x+\alpha (mod 1) by an irrational number \alpha on the additive circle T=[0,1). Let \beta_1,..., \beta_d be d\geqslant 1 parameters in [0, 1). One of the goals of this paper is to describe the…

The origin of deterministic diffusion is a matter of discussion. We study the asymptotic distributions of the sums $y_n(x)=\sum_{k=0}^{n-1}\psi (x+k\alpha)$, where $\psi$ is a periodic function of bounded variation and $\alpha$ an…

We study the fluctuations of ergodic sums using global and local specifications on periodic points. We obtain Lindeberg-type central limit theorems in both situations. As an application, when the system possesses a unique measure of maximal…

In this paper we study the ergodic properties of mathematical billiards describing the uniform motion of a point in a flat torus from which finitely many, pairwise disjoint, tubular neighborhoods of translated subtori (the so called…

We establish a connection between the structure of a stationary symmetric alpha-stable random field (0 < alpha < 2) and ergodic theory of non-singular group actions, elaborating on a previous work by Rosinski (2000). With the help of this…

We introduce the concepts of rotation numbers and rotation vectors for billiard maps. Our approach is based on the birkhoff ergodic theorem. We anticipate that it will be useful, in particular, for the purpose of establishing the…

Deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of modulus 1. We consider the corresponding eigenfunctions, and in Theorem 1.1 we prove that the limit inferior of the ergodic…

We demonstrate for a generic pseudointegrable billiard that the number of periodic orbit families with length less than $l$ increases as $\pi b_0l^2/\langle a(l) \rangle$, where $b_0$ is a constant and $\langle a(l) \rangle$ is the average…

We prove polynomial upper bounds for the deviation of ergodic averages for the straight line flow on every translation surface in almost every direction, in particular for those surfaces arising from rational polygonal billiards.

We consider classical billiards in plane, connected, but not necessarily bounded domains. The charged billiard ball is immersed in a homogeneous, stationary magnetic field perpendicular to the plane. The part of dynamics which is not…

We study the size of the set of ergodic directions for the directional billiard flows on the infinite band $\R\times [0,h]$ with periodically placed linear barriers of length $0<\lambda<h$. We prove that the set of ergodic directions is…

Rotations on the circle by irrational numbers give rise to uniquely ergodic Sturm dynamical systems. We show that rotations by badly approximable irrationals have the property of fast ergodicity. It was shown recently that any Sturmian…

We consider dynamics of scalar semilinear parabolic equations on bounded intervals with periodic boundary conditions, and on the entire real line, with a general nonlinearity $g(t,x,u,u_x)$ either not depending on $t$, or periodic in $t$.…

We consider a class of semi-linear differential Volterra equations with memory terms, polynomial nonlinearities and random perturbation. For a broad class of nonlinearities, we study statistically steady states of the system and find that…