Exponential decay of correlations in multi-dimensional dispersing billiards
type: Item
instance of: scholarly article
title: Exponential decay of correlations in multi-dimensional dispersing billiards
zbMATH Open document ID: 1162.37018
author: Péter Bálint
author: Imre Péter Tóth
published in: Annales Henri Poincaré
publication date: 2009-02-26T00:00:00Z
review text: The setting for this paper is a connected billiards domain \(Q\) in the \(d\)-dimensional flat torus \(\mathbb{T}^d\). A point particle travels uniformly (along straight lines with constant speed) within \(Q\) and bounces off the boundary \(\partial Q\) in elastic collisions. The boundary \(\partial Q\) is assumed to be a finite collection of \(d-1\)-dimensional J\(C^3\)-smooth submanifolds of \(\mathbb{T}^d\). The billiard system is taken to be strictly dispersing (the second fundamental form, or curvature operator, is positive definite on \(\partial Q\)), the billiard horizon is finite (the distance along a straight line beginning in \(\partial Q\) in a specified direction to the point where the trajectory reaches \(\partial Q\) again is uniformly bounded for all starting points and directions), and their are no corners on the boundary. Finally, the authors make a critical technical assumption that the complexity of the singularity set grows subexponentially with the number of iterates of the billiards map. Under these assumptions, the authors prove exponential decay of correlations. The authors' approach depends on a detailed local analysis. In a companion paper [Isr. J. Math. 167, 155--175 (2008; Zbl 1162.37004)], \textit{P. Bachurin}, \textit{P. Bálint} and \textit{I. P. Tóth} prove that this class of billiard systems is also ergodic.
reviewed by: William J. Satzer Jr.
Mathematics Subject Classification ID: 37D50
Mathematics Subject Classification ID: 82C40
Mathematics Subject Classification ID: 37A60
zbMATH DE Number: 5519124
zbMATH Keywords: dispersing billiards
zbMATH Keywords: correlations
zbMATH Keywords: ergodicity singularly manifold
zbMATH Keywords: subexponential complexity
MaRDI profile type: Publication
full work available at URL: s00023-008-0389-1
OpenAlex ID: W2037412506
DOI: 10.1007/S00023-008-0389-1
Recommended article: Dispersing billiards with moving scatterers
Recommended article: Proving the ergodic hypothesis for billiards with disjoint cylindric scatterers
Recommended article: Many-dimensional nowhere dispersing billiards with chaotic behavior
Recommended article: Billiards with positive topological entropy
Recommended article: Q2709109
Recommended article: On statistical properties of hyperbolic systems with singularities
Recommended article: Decay of correlations and dispersing billiards.
Recommended article: Upgrading the local ergodic theorem for planar semi-dispersing billiards
Recommended article: Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows
Recommended article: Q2997836
description: scientific article; zbMATH DE number 5519124