Statistical properties of actions of periodic orbits

Abstract

We investigate statistical properties of unstable periodic orbits, especially actions for two simple linear maps (p-adic Baker map and sawtooth map). The action of periodic orbits for both maps is written in terms of symbolic dynamics. As a result, the expression of action for both maps becomes a Hamiltonian of one-dimensional spin systems with the exponential-type pair interaction. Numerical work is done for enumerating periodic orbits. It is shown that after symmetry reduction, the dyadic Baker map is close to generic systems, and the p-adic Baker map and sawtooth map with noninteger K are also close to generic systems. For the dyadic Baker map, the trace of the quantum time-evolution operator is semiclassically evaluated by employing the method of Phys. Rev. E 49, R963 (1994). Finally, using the result of this and with a mathematical tool, it is shown that, indeed, the actions of the periodic orbits for the dyadic Baker map with symmetry reduction obey the uniform distribution modulo 1 asymptotically as the period goes to infinity. (c) 2000 American Institute of Physics.

Periodic orbit action correlations in the Baker map

“…4d for n = 65. This oscillatory behavior has already been observed by Sano (1999) when studying periodic orbit correlations in the Baker map without separating the two symmetry subspaces. Sano's result can now be interpreted in terms of the spectrum of the quasiclassical operator (23); the exponents γ ± 0 for the two different subspaces, which determine the asymptotic behavior of P (s, n) for large n, see Eq.…”

“…periodic orbit sums, considerably. Other quasiclassical operators for the Baker map have been proposed by Kaplan and Heller (1996), Sano (1999) and Hannay (1999). Note, that an identification of semiclassical periodic orbit sums with classical or quasiclassical operator is not possible for generic systems and the notation TrŨ n is then a mere substitution for the periodic orbit sum itself.…”

“…Argaman et al (1993) present numerical results which agree qualitatively with the RMT-prediction presented in the same paper; Cohen et al (1998) find correlations in the length spectrum of periodic orbits in the Stadium billiard which indicate RMT-like correlations but a quantitative analysis can not compete with the accuracy reached for spectral eigenvalue statistics. Other groups report complete failure in their search for any kind of action correlations (Aurich 1998) or find action correlations which do not coincide with the RMT -prediction by Argaman et al (1993), see Sano (1999); in addition there are yet no arguments other than from the duality between quantum eigenvalues and classical periodic orbits which indicate why universal classical correlations should exist. Cohen et al (1998) present a variety of ideas based on identifying relationships within various subgroups of orbits in the Sinai billiard.…”

Action correlations and random matrix theory

“…Second, to test the general semiclassical arguments on action correlations for a paradigm chaotic dynamical system -the Baker map. This system was investigated previously by a number of groups, [2,3,6,8], who demonstrated numerically the existence of the expected correlations. Here, we develop another approach for the analysis of the action spectrum, where we try to systematically asses the way the periodic orbits and their actions can be partitioned to families which are dynamically related.…”

“…were the universality of action correlations, and their relation to Random Matrix Theory (RMT) were studied for a few chaotic systems [2]. Various aspects of the subject were investigated later [3,4,6,7,8,9,14]. This culminated recently in the work of Sieber and Richter [11], who identified pairs of correlated trajectories, whose contribution to the spectral form factor for systems with time reversal symmetry is identical to the next to leading order term in the formfactor predicted by RMT (see also [12,13]).…”