Stability ordering of cycle expansions

may be made to only a short time of order l 21 1 in the future, where l1 is the largest Lyapunov exponent. Despite this unpredictability, much can be said about the average behavior of the system in many cases using only a small number of unstable periodic orbits or “cycles.” This can be achieved by using cycle expansions [1,2] of Ruelle’s dynamical zeta function [3]. Cycle expansions have proved very useful in both classical and quantum chaos, giving accurate estimates of the escape rate of open billiard systems [4] and the energy levels of helium [5] using a surprisingly small number of classical cycles. The main idea of this approach is that a long generic trajectory may be approximated by various periodic orbits at different times, and that longer periodic orbits may often be “shadowed” by shorter “fundamental” cycles, closely following the shorter cycles along different sections of its length. Thus averages are calculated using fundamental cycles, with small corrections due to longer cycles. Periodic orbits which are exact repetitions of smaller cycles are explicitly summed, so that all expressions are written in terms of the remaining “prime” cycles. The expansions work best when the symbolic dynamics is well understood, and long periodic orbits are well shadowed by shorter ones. In this paper we investigate a system in which neither of these conditions holds, the strong-field Lorentz gas. In spite of these difficulties, reasonable results may be obtained by ordering the expansion in terms of stability rather than the length of periodic orbits. This approach should be valid wherever cycle expansions can be applied, including flows for which a natural topological “length” is difficult to define. The origin of cycle expansions is a well developed theory of trace formulas and dynamical zeta functions [3,6]. Here we will need only the expression for the classical time average of some quantity A in a closed system [2],