Specific heat of multifractal energy spectra

Motivated by the self-similar character of energy spectra demonstrated for quasicrystals, we investigate the case of multifractal energy spectra, and compute the specific heat associated with simple archetypal forms of multifractal sets as generated by iterated maps. We considered the logistic map and the circle map at their threshold to chaos. Both examples show nontrivial structures associated with the scaling properties of their respective chaotic attractors. The specific heat displays generically log-periodic oscillations around a value that characterizes a single exponent, the "fractal dimension," of the distribution of energy levels close to the minimum value set to 0. It is shown that when the fractal dimension and the frequency of log oscillations of the density of states are large, the amplitude of the resulting log oscillation in the specific heat becomes much smaller than the log-periodic oscillation measured on the density of states.