We consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant, this measure is exact-dimensional and the almost everywhere the local scaling exponent is a smooth function of the parameter, is strictly smaller than the Hausdorff dimension of the spectrum, and converges to one as the coupling constant 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 invariant surface (level surface of the Fricke-Vogt invariant). This allows us to make a connection between the spectral problem at hand and the dimension theory of dynamical systems.
This is a joint work with David Damanik.