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.