Clustering of periodic orbits in chaotic systems

In the framework of the semiclassical approach, the universal spectral correlations in Hamiltonian systems with classical chaotic dynamics can be attributed to the systematic correlations between the actions of periodic orbits which (up to the switch in the momentum direction) pass through approximatey the same points of the phase space. By considering symbolic dynamics of the system one can introduce a natural ultrametric distance between periodic orbits and organize them into clusters of orbits approaching each other in the phase space. We study the distribution of cluster sizes for the baker fs map in the asymptotic limit of long trajectories. This problem is equivalent to the one of counting degeneracies in the length spectrum of the de Bruijn graphs. Based on this fact, we derive the probability Pk that k randomly chosen periodic orbits belong to the same cluster. Furthermore, we find asymptotic behaviour of the largest cluster size |C_max| and derive the probability P _(t) that a random periodic orbit belongs to a cluster smaller than t C_max|, t ε [0, 1].