Decay of correlations and control of chaotic billiards

We consider a two-dimensional square Sinaï billiard with a centered disc, a computer simulation of which shows that a specific correlation function displays an initial nonexponential decay. The initial nonexponential era is larger when the Lyapounov exponent is smaller. The onset of the exponential era corresponds to the onset of chaos in the system, and the initial nonexponential era can be understood as the preparation time for the manifestation of chaos. By suitable perturbation of the radius of the central disc, the initial nonexponential era increases 8 times in length, thus giving rise to a particular parametric control of the system.