For locally constant cocycles defined on an aperiodic subshift, Damanik and Lenz (Duke Math J 133(1): 95-123, 2006) proved that if the subshift satisfies a certain condition (B), then the cocycle is uniform. In this paper, we study simple Toeplitz subshifts. We give a criterion that simple Toeplitz subshifts satisfy condition (B), and also give some sufficient conditions that they do not satisfy condition (B). However, we can still prove the uniformity of Schrödinger cocycles over any simple Toeplitz subshift. As a consequence, the related Schrödinger operators have Cantor spectrum of Lebesgue measure 0. We also exhibit a fine structure for the spectrum, and this helps us to prove purely singular continuous spectrum for a large class of simple Toeplitz potentials.

Abstract. Let α ∈ (0, 1) be irrational and [0; a1, a2, · · · ] be the continued fraction expansion of α. Let Hα,V be the Sturm Hamiltonian with frequency α and coupling V , Σα,V be the spectrum of Hα,V . The fractal dimensions of the spectrum have been determined by Fan, Liu and Wen (Erg. Th. Dyn. Sys.,2011) when {an} n≥1 is bounded. The present paper will treat the most difficult case, i.e, {an} n≥1 is unbounded. We prove that for V ≥ 24, dimH Σα,V = s * (V ) and dimB Σα,V = s * (V ), where s * (V ) and s * (V ) are lower and upper pre-dimensions respectively. By this result, we determine the fractal dimensions of the spectrums for all Sturm Hamiltonians.We also show the following results: s * (V ) and s * (V ) are Lipschitz continuous on any bounded interval of [24, ∞); the limits s * (V ) ln V and s * (V ) ln V exist as V tend to infinity, and the limits are constants only depending on α; s * (V ) = 1 if and only if lim sup n→∞ (a1 · · · an) 1/n = ∞, which can be compared with the fact: s * (V ) = 1 if and only if lim infn→∞(a1 · · · an) 1/n = ∞( Liu and Wen, Potential anal. 2004).

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Abstract. 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 Hölder 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.

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost additive continuous potentials (φn) ∞ n=1 on a topologically mixing subshift of finite type X endowed itself with a metric associated with such a potential. We work without additional regularity assumption other than continuity. Our approach differs from those used previously to deal with this question under stronger assumptions on the potentials. As a consequence, it provides a new description of the structure of the spectrum in terms of weak concavity. Also, the lower bound for the spectrum is obtained as a consequence of the study sets of points at which the asymptotic behavior of φn(x) is localized, i.e. depends on the point x rather than being equal to a constant. Specifically, we compute the Hausdorff dimension of sets of the form {x ∈ X : limn→∞ φn(x)/n = ξ(x)}, where ξ is a given continuous function. This has interesting geometric applications to fixed points in the asymptotic average for dynamical systems in R d , as well as the fine local behavior of the harmonic measure on conformal planar Cantor sets.it is ready to show that χ is also injective. Next we show that T (Σ, which is a contradiction. Next we show that for any Gibbs measure µ we have µ(Z ∞ ) = 0. Define Z n := w∈Σ A, * , |w|≤n f w (∂V ) and Z n = χ −1 ( Z n ). The sequence (Z n ) n≥1 is non decreasing and Z ∞ = n≥1 Z n . Since the IFS is conformal we can easily get T (Z n ) ⊂ Z n−1 for n ≥ 1 and T (Z 0 ) ⊂ Z 0 . Consequently T (Z n ) ⊂ Z n . By the ergodicity we have µ(Z n ) = 0 or 1. By the SOSC, Σ A \Z n is nonempty and open, thus by the Gibbs property of µ we get µ(Σ A \ Z n ) > 0, hence µ(Z n ) = 0.Proof of Theorem 3.2. (1) At first we notice that by the property (4) assumed in the construction of J the mapping χ is Lipschitz. This is enough to get the desired upper bounds from Theorem 2.2(1). Now we deal with the lower bound for dimensions and the equality L Φ = L Φ . We notice that the inclusion L Φ ⊂ L Φ holds by construction.Suppose J is a conformal repeller. Since we have χ • T = g • χ on Σ A and χ is surjective, it is seen that χ −1 (E Φ (α)) = E Φ (α) for any α ∈ L Φ . Thus L Φ = L Φ and by Proposition 5, we have dim H E Φ (α) = dim H E Φ (α).Suppose J is the attractor of a conformal IFS with SOSC. Let α ∈ L Φ . Let Z = χ −1 (∂V ). The set Z is closed and by Lemma 3.