  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 $d_V$ 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.
