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 Hausdor 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.