Periodic orbits and chaotic-diffusion probability distributions

Periodic-orbit (PO) formulas for chaotic-diffusion probability distributions (PDs) are examined in the case of the perturbed Arnolʹd cat map on the cylinder. This translationally invariant system exhibits a transition from uniform to nonuniform hyperbolicity as the perturbation parameter is increased. Two coarse-grained PDs, describing the “diffusion” between unit cells of the system, are studied: (a) a PD based on PO ensembles; (b) a PD based on generic ensembles. The approximate PO formula for PD (b) gives results which fluctuate around the expected Gaussian distribution for all parameters considered and thus agree qualitatively with results from standard methods. The exact PO formula for PD (a) gives similar results only for sufficiently small parameters. The results for large parameters decrease monotonically relative to the Gaussian distribution. This deviation seems to disappear as the PO period is increased.