The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian

Abstract

We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant V, this measure is exact-dimensional and the almost everywhere value dV of the local scaling exponent is a smooth function of V, is strictly smaller than the Hausdorff dimension of the spectrum, and converges to one as V tends to zero. The proof relies on a new connection between the density of states measure of the Fibonacci Hamiltonian and the measure of maximal entropy for the Fibonacci trace map on the non-wandering set in the V-dependent invariant surface. This allows us to make a connection between the spectral problem at hand and the dimension theory of dynamical systems.