Exponential decay of correlations in multi-dimensional dispersing billiards
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.
scientific article; zbMATH DE number 5519124