Statistics of energy levels in Hamiltonian systems with classically
chaotic dynamics belong to one of three Wigner-Dyson symmetry classes. By
the semiclassical approach this remarkable universality can be attributed
to the systematic correlations between actions of periodic orbits. The
correlating periodic orbits do not exist as independent individuals
but rather come in closely packed bunches. We demonstrate how this
bunching phenomena can be rigorously described using symbolic
dynamics of the system.
For systems with Markov partition we introduce an ultrametric
distance between periodic orbits and organize them into clusters.
Each cluster is composed of periodic orbits passing through
(approximately) the same points of the phase space. We then study the
distribution of cluster sizes in the asymptotic limit of long
trajectories. In the case of the baker's map this problem turns out to be
equivalent to the one of counting degeneracies in the length spectrum
of de Bruijn graphs.