Multidimensional hyperbolic billiards

Abstract

Dedicated to the memory of Kolya Chernov.The theory of hyperbolic billiards having been created by Sinai, Kolya did more than anyone else to found their multidimensional theory.Abstract. The theory of planar hyperbolic billiards is already quite well developed by having also achieved spectacular successes. In addition there also exists an excellent monograph by Chernov and Markarian on the topic. In contrast, apart from a series of works culminating in Simányi's remarkable result on the ergodicity of hard ball systems and other sporadic successes, the theory of hyperbolic billiards in dimension 3 or more is much less understood. The goal of this work is to survey the key results of their theory and highlight some central problems which deserve particular attention and efforts.

Three-dimensional billiards: Visualization of regular structures and trapping of chaotic trajectories

“…Three-dimensional (3d) billiards (see the upper right inset in Fig. 1 for an illustration) have in particular been investigated for establishing fully chaotic dynamics [17][18][19][20][21][22][23][24][25][26][27], and studying both classical and quantum properties of integrable, mixed and fully chaotic systems, see e.g. [28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43].…”

Rates of convergence for the superdiffusion in the Boltzmann–Grad limit of the periodic Lorentz gas

“…The central limit theorem for the Lorentz gas with infinite horizon in high dimension remains an open problem. We refer the readers to [8,26,10] for a description of the diffusive properties of the Lorentz gas. Recently, Lutsko-Toth [19] derived the invariance principle for the random Lorentz gas, under both the Boltzmann-Grad limit and the simultaneous long time limit.…”

Inducing schemes for multi-dimensional piecewise expanding maps

“…Furthermore, our assumptions on the discontinuities are weaker than those of [3], which may provide some insight in generalizing [3] to cover multidimensional billiards. We refer the reader to the survey [6] for the relevance of piecewise expanding maps in providing insights into the problems surrounding the study of multi-dimensional billiards. Other systems to which the results of the current paper could possibly be applied (in my opinion) are variations on "Hu-Vaienti" maps [5] and "Viana" maps [7,1].…”