Universality and the power spectrum at the onset of chaos

Two one-dimensional maps are iterated to evaluate the average height φ(k) of the peaks in the power spectrum corresponding to frequencies ω _k,l -(2/-1)π/2 ^k , where /-1.2 … 2 ^{k −1} and k-1,2, … at the onset of chaos. It is shown that the ratio ϕ(k)/ϕ(k +1) is nearly constant and for large k approaches a universal limit 2ß ⁽²⁾-20.963.…