Related papers: Proof of the Ergodic Hypothesis for Typical Hard B…

We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (``cat maps''). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the…

For a general radially symmetric, non-increasing, non-negative kernel $h\in L ^ 1 _{loc} ( R ^ d)$, we study the rigidity of measurable sets in $R ^ d$ with constant nonlocal $h$-mean curvature. Under a suitable "improved integrability"…

The Northeast Model is a spin system on the two-dimensional integer lattice that evolves according to the following rule: Whenever a site's southerly and westerly nearest neighbors have spin $1$, it may reset its own spin by tossing a…

We first construct nonholonomic systems of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$ with centers $O_1,...,O_n$ and with the same radius $r$ that are rolling without slipping around a fixed sphere $\mathbf S_0$ with center $O$…

We study a natural class of Fermi-Ulam Models that features good hyperbolicity properties and that we call dispersing Fermi-Ulam models. Using tools inspired by the theory of hyperbolic billiards we prove, under very mild complexity…

We prove ergodicity of a class of infinite measure preserving systems, called skew-products. More precisely, we consider systems of the form \[ {T_f}:{[0, 1) \times \mathbb{R}}\to{[0, 1) \times \mathbb{R}},\quad {T_f(x, t)}:={(T(x),…

We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain…

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}…

We study a stability property of probability laws with respect to small violations of algorithmic randomness. A sufficient condition of stability is presented in terms of Schnorr tests of algorithmic randomness. Most probability laws, like…

In the present paper we prove a strong form of Arnold diffusion. Let $\mathbb{T}^2$ be the two torus and $B^2$ be the unit ball around the origin in $\mathbb{R}^2$. Fix $\rho>0$. Our main result says that for a "generic" time-periodic…

In this paper we mainly study the dynamical complexity of Birkhoff ergodic average under the simultaneous observation of any number of continuous functions. These results can be as generalizations of [6,35] etc. to study Birkhorff ergodic…

We investigate a three-parameter family of billiard tables with circular arc boundaries. These umbrella-shaped billiards may be viewed as a generalization of two-parameter moon and asymmetric lemon billiards, in which the latter classes…

Let $G$ be a semi-direct product $G=A\times_\phi K$ with $A$ Abelian and $K$ compact. We characterize spread-out probability measures on $G$ that are mixing by convolutions by means of their Fourier transforms. A key tool is a spectral…

The paper provides a systematic characterization of quantum ergodic and mixing channels in finite dimensions and a discussion of their structural properties. In particular, we discuss ergodicity in the general case where the fixed point of…

We consider the volume of a Boolean expression of some congruent balls about a given system of centers in the $d$-dimensional Euclidean space. When the radius $r$ of the balls is large, this volume can be approximated by a polynomial of…

Let $X$ be a compact complex surface. Consider a finitely supported probability measure $\mu$ on $\text{Aut}(X)$ such that $\Gamma_{\mu} = \langle \text{Supp}(\mu)\rangle<\text{Aut}(X)$ is non-elementary. We do not assume that…

In a previous paper of Winter and the author the Law of Large Numbers for the local mass of certain superdiffusions was proved under a spectral theoretical assumption, which is equivalent to the ergodicity (positive recurrence) of the…

Although the G.Birkhoff Ergodic Theorem (BET) is trivial for finite spaces, this does not help in proving it for hyperfinite Loeb spaces. The proof of the BET for this case, suggested by T. Kamae, works, actually, for arbitrary probability…

When a granular system is tapped, its volume changes. Here, using a well-defined macroscopic protocol, we prepare an ensemble of granular systems and track the statistics of volume changes as a function of the number of taps. This is in…

We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversaly conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudo-group.…