Self-similar correlations in a barrier billiard

We give a renormalization analysis of the self-similarity of autocorrelation functions in symmetric barrier billiards for golden mean trajectories. For the special case of a half-barrier we present a rigorous calculation of the asymptotic height of the main peaks in the autocorrelation function. Fundamental to this work is a detailed analysis of a functional recurrence equation which has previously been used in the analysis of fluctuations in the Harper equation and of correlations in strange non-chaotic attractors and in quantum two-level systems.