Approximation and Billiards

1 9 N ov 2 00 7 APPROXIMATION AND BILLIARDS

This survey is based on a series of talks I gave at the conference " Dynamical systems and Diophantine approximation " at l'Instut Henri Poincaré in June 2003. I will present asymptotic results (transitivity, ergodicity, weak-mixing) for billiards based on the approximation technique developed by Katok and Zemlyakov. I will also present approximation techniques which allow to prove the abundanc...

N ov 2 00 6 APPROXIMATION AND BILLIARDS

This survey is based on a series of talks I gave at the conference " Dynamical systems and diophantine approximation " at l'Instut Henri Poincaré in June 2003. I will present asymptotic results (transitivity, ergodicity, weak-mixing) for billiards based on the approximation technique developed by Katok and Zemlyakov. I will also present approximation techniques which allow to prove the abundanc...

Approximating multi-dimensional Hamiltonian flows by billiards

The behavior of a point particle traveling with a constant speed in a region D ∈ RN , undergoing elastic collisions at the regions’s boundary, is known as the billiard problem. Various billiard models serve as approximation to the classical and semi-classical motion in systems with steep potentials (e.g. for studying classical molecular dynamics, cold atom’s motion in dark optical traps and mic...

Inhomogeneous Diophantine Approximation And Angular Recurrence For Polygonal Billiards

Trace Formulas and Spectral Statistics of Diffractive Systems

Diffractive systems are quantum-mechanical models with point-like singularities where usual semiclassical approximation breaks down. An overview of recent investigations of such systems is presented. The following examples are considered in details: (i) billiards (both inte-grable and chaotic) with small-size scatterers, (ii) pseudo-integrable polygonal plane billiards, and (iii) billiards with...