Spectral statistics of 'cellular' billiards

For a bounded domain Ω0 subset of R2 whose boundary contains a number of flat pieces Γi, i = 1, ..., l we consider a family of non-symmetric billiards Ω constructed by patching several copies of Ω0 along Γis. It is demonstrated that the length spectrum of the periodic orbits in Ω is degenerate with the multiplicities determined by a matrix group G. We study the energy spectrum of the corresponding quantum billiard problem in Ω and show that it can be split into a number of uncorrelated subspectra corresponding to a set of irreducible representations α of G. Assuming that the classical dynamics in Ω0 are chaotic, we derive a semiclassical trace formula for each spectral component and show that their energy level statistics are the same as in standard random matrix ensembles. Depending on whether α is real, pseudo-real or complex, the spectrum has either Gaussian orthogonal, Gaussian symplectic or Gaussian unitary types of statistics, respectively