Superconductor-proximity effect in hybrid structures: Fractality versus Chaos

We study the proximity effect of a superconductor to a normal system with fractal spectrum. We find that there is no gap in the excitation spectrum, even in the case where the underlying classical dynamics of the normal system is chaotic. An analytical expression for the distribution of the smallest excitation eigenvalue $E_g$ of the hybrid structure is obtained.On small scales it decays algebraically as $\mathcal{P}(E_g)\sim E_g^{-D_0}$, where $D_0$ is the fractal dimension of the spectrum of the normal system. Our theoretical predictions are verified by numerical calculations performed for various models.