0.1. Preface. In 1963 Ya. G. Sinai [Sin(1963)] formulated a modern version of Boltzmann’s ergodic hypothesis, what we now call the “Boltzmann-Sinai Ergodic Hypothesis”: The billiard system of N (N ≥ 2) hard balls of unit mass moving on the flat torus T = R/Z (ν ≥ 2) is ergodic after we make the standard reductions by fixing the values of trivial invariant quantities. It took fifty years and the...

We prove a quantitative rate of homogenization for the G equation in random environment with finite range dependence. Using ideas from percolation theory, proof bootstraps result Cardaliaguet–Souganidis, who proved qualitative more general ergodic environment.

Abstract Let $\Gamma $ be a Zariski dense Anosov subgroup of connected semisimple real algebraic group $G$. For maximal horospherical $N$ $G$, we show that the space all non-trivial $NM$-invariant ergodic and $A$-quasi-invariant Radon measures on \backslash G$, up to proportionality, is homeomorphic ${\mathbb {R}}^{\text {rank}\,G-1}$, where $A$ split torus $M$ compact normalizes $N$. One main ...

Abstract In this paper, we study the ergodicity of geodesic flows on surfaces with no focal points. Let M be a smooth connected and closed surface equipped $C^{\infty }$ Riemannian metric g , whose genus $\mathfrak {g} \geq 2$ . Suppose that $(M,g)$ has We prove flow unit tangent bundle is ergodic respect to Liouville measure, under assumption set points negative curvature at most finitely many...

The Bessel process models the local eigenvalue statistics near 0 of certain large positive definite matrices. In this work, we consider probability P ∃ no points in on ( , x 1 ) ∪ 2 3 ⋯ g + $$\begin{equation*} {\mathbb{P}\left(\exists \text{ on}\ (0,{x}_{1})\cup ({x}_{2},{x}_{3})\cup \cdots \cup ({x}_{2g},{x}_{2g+1})\right),} \end{equation*}$$ where < $0<x_{1}<\cdots <x_{2g+1}$ and ≥ $g \ge 0$ ...

We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure μ on a countable group G, A. Nevo showed that a necessary condition for G to have property (T) is that the Furstenberg μ-entropy values of the ergodic, properly nonsingular G-actions are bounded away from zero. We show that this is also a sufficient condit...

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Γ a closed subgroup of G. Let W be a subgroup of G such that AdG(W ) is contained in the Zariski clos...

This paper is a continuation of the authors' previous work on noncommutative joinings, and contains study relative independence W$^*$-dynamical systems. We prove that, given any separable locally compact group $G$, an ergodic W$^{*}$-dynamical $G$-system $\mathfrak{M}$ with subsystem $\mathfrak{N}$ disjoint to from its maximal $\mathfrak{M}_{K}$ if only $\mathfrak{N}\cong\mathfrak{M}_{K}$. gene...

In this paper, we prove a convergence theorem for singular perturbations problems class of fully nonlinear parabolic partial differential equations (PDEs) with ergodic structures. The limit function is represented as the viscosity solution to degenerate PDEs. Our approach mainly based on G-stochastic analysis argument. As byproduct, also establish averaging principle stochastic driven by G-Brow...

In this paper we introduce and discuss various notions of doubling for measurepreserving actions a countable abelian group G. Our main result characterizes 2-doubling actions, can be viewed as an ergodic-theoretical extension some classical density theorems sumsets by Kneser. All our results are completely sharp they new already in the case when G = (Z; +).