  We study the spectral properties of the Sturm Hamiltolian of eventually
constant type, which includes the Fibonacci Hamiltonian. Let $s$ be the
Hausdorff dimension of the spectrum. For $V>20$, we show that the restriction
of the $s$-dimensional Hausdorff measure to the spectrum is a Gibbs type
measure; the density of states measure is a Markov measure. Based on the fine
structures of these measures, we show that both measures are exact dimensional;
we obtain exact asymptotic behaviors for the optimal H\"older exponent and the
Hausdorff dimension of the density of states measure and for the Hausdorff
dimension of the spectrum. As a consequence, if the frequency is not silver
number type, then for $V$ big enough, we establish strict inequalities between
these three spectral characteristics.
  We achieve them by introducing an auxiliary symbolic dynamical system and
applying the thermodynamical and multifractal formalisms of almost additive
potentials.
