Specific heat oscillations in quasi-periodic structures

Most models on quasi-crystals, as well as the related experimental results, exhibit fractal energy spectra. We discuss the relevant thermodynamic implications of this feature by performing both analytical and numerical calculations of the specific heats associated with successive hierarchical approximations to fractal energy spectra. Instructive anomalies are displayed, namely the specific heat exhibits an infinite number of small-amplitude oscillations symmetrically disposed around a characteristic dimension of the spectrum. We also analyse a multifractal spectrum generated dynamically by the logistic map and conjecture a possible connection with Tsallisʹ generalized thermostatics.