Dynamical Lifshitz Tails

Abstract

We consider one-parameter families of random circle diffeomorphisms $g_{E,y}$ for which the unperturbed map $g_{0,\bar{0}}$ has a fixed point of order $2k$ and the dependence on the parameter $E$ is monotone. Under reasonable assumptions, we show that the rotation number $ρ(E)$ exhibits Lifshitz tail decay with exponent $-\frac{2k - 1}{2k}$, \[ \lim_{E \downarrow 0} \frac{\ln(-\ln(ρ(E) - ρ(0)))}{\ln(E)} = -\frac{2k-1}{2k}. \] The exponent is determined by the passage time through a parabolic bottleneck. A full rotation requires on the order of $E^{-\frac{2k - 1}{2k}}$ successive small perturbations, and the probability of such a streak decays exponentially as a function of its length. When $k=1$, the exponent is $-1/2$, and we recover as a corollary a purely dynamical proof of Lifshitz tail asymptotics at the spectral edges of the one-dimensional Anderson model.