The Hausdorff Dimension of the Spectrum of a Class of Generalized Thue-Morse Hamiltonians

Let τ be a generalized Thue-Morse substitution on a two-letter alphabet {a, b}: τ (a) = a ^m b ^m, τ(b) = b ^m a ^m for the integer m ≥ 2. Let ξ be a sequence in {a, b}^ℤ that is generated by τ. We study the one-dimensional Schrödinger operator H_m,λ on l ² (ℤ) with a potential given by v(n) = λV_ξ(n) , where λ > 0 is the coupling and V_ξ(n) = 1 (V_ξ(n) = −1) if ξ(n) = a (ξ(n) = b). Let Λ₂ = 2, and for m > 2, let Λ _m = m if m = 0 mod 4; let Λ _m = m − 3 if m ≡ 1 mod 4; let Λ _m = m − 2 if m ≡ 2 mod 4; let Λ _m = m − 1 if m ≡ 3 mod 4. We show that the Hausdorff dimension of the spectrum σ(H_m,λ) satisfies that dimHσ(Hm,λ)>logΛmlog64m+4. It is interesting to see that dim _H σ(H_m,λ) tends to 1 as m tends to infinity.