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.