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

The spectral properties of the strongly coupled Sturm Hamiltonian of eventually constant type