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Chaotic billiards generated by arithmetic groups. (English) Zbl 0968.81514

Summary: It is known that statistical properties of the energy levels for various billiards on a constant-negative-curvature surface do not follow the universal random-matrix predictions. We show that nongeneric behavior of the systems investigated so far originates from the special arithmetic nature of their tiling groups, which produces an exponentially large degeneracy of lengths of periodic orbits. A semiclassical study of the two-point correlation function shows that the spectral fluctuations are close to Poisson-like ones, typical of integrable systems.

MSC:

81Q50 Quantum chaos
20H15 Other geometric groups, including crystallographic groups
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
82B05 Classical equilibrium statistical mechanics (general)
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