Spatially chaotic configurations and functional equations with rescaling

The functional equation y(qt) = 1/4q[y(t + 1) + y(t - 1) + 2y(t)] (0 < q < 1) t ∈ ℝ is associated with the appearance of spatially chaotic structures in amorphous (glassy) materials. Continuous compactly supported solutions of the above equation are of special interest. We shall show that there are no such solutions for 0 < q < 1/2, whereas such a solution exists for almost all 1/2 < q < 1. The words 'for almost all q' in the previous sentence cannot be omitted. There are exceptional values of q in the interval [1/2, 1] for which there are no integrable solutions. For example, q = (√5 - 1)/2 ≈ 0.618, which is the reciprocal of the 'golden ratio' is such an exceptional value. More generally, if λ is any Pisot-Vijayaraghavan number, or any Salem number, then q = λ^-1 is an exceptional value.