The autocorrelation function of a pseudointegrable system

The autocorrelation function of a pseudointegrable system is considered. The system consists of “billiards” on plane surface formed out of three squares arranged in an “L” shape. This system has the important property of being constructed from copies of an integrable subsystem, the single square. The motion can be decomposed into a continuous and a discrete part, the unpredictability in the system being associated with the latter. A discrete autocorrelation function is calculated, and its decay properties investigated. Structure found in this autocorrelation function is associated with the continued fraction expansion of the ratio of velocity components. For repeating continued fractions, such as the golden mean, the autocorrelation function exhibits a selfsimilar structure. For the general case of a randomly chosen velocity ratio, we derive the time dependence of the number of occurences of “large” autocorrelation values, which differs from the behavior in integrable and chaotic systems.