We study the spectral properties of discrete Schrödinger operators with potentials given by primitive invertible substitution sequences (or by Sturmian sequences whose rotation angle has an eventually periodic continued fraction expansion, a strictly larger class than primitive invertible substitution sequences). It is known that operators from this family have spectra which are Cantor sets of zero Lebesgue measure. We show that the Hausdorff dimension of this set tends to 1 as coupling constant λ tends to 0. Moreover, we also show that at small coupling constant, all gaps allowed by the gap labeling theorem are open and furthermore open linearly with respect to λ. Additionally, we show that, in the small coupling regime, the density of states measure for an operator in this family is exact dimensional. The dimension of the density of states measure is strictly smaller than the Hausdorff dimension of the spectrum and tends to 1 as λ tends to 0.

Abstract. We consider a family of discrete Jacobi operators on the onedimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schrödinger operators, although some of the results that we obtain differ qualitatively and quantitatively from those for the Schrödinger operators. The nontrivialities of this extension lie in the dynamics of the associated trace map as one attempts to extend the trace map formalism from the Schrödinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are, in a sense, a test item, as many other models can be attacked via the same techniques, and we present an extensive discussion on this.

ABSTRACT. An augmented generalized happy function, S [c,b] maps a positive integer to the sum of the squares of its base-b digits and a non-negative integer c. A positive integer u is in a cycle of S [c,b] if, for some positive integer k, S k [c,b] (u) = u and for positive integers v and w, v is w-attracted for S [c,b] if, for some non-negative integer ℓ, S ℓ [c,b] (v) = w. In this paper, we prove that for each c ≥ 0 and b ≥ 2, and for any u in a cycle of S [c,b] , (1) if b is even, then there exist arbitrarily long sequences of consecutive u-attracted integers and (2) if b is odd, then there exist arbitrarily long sequences of 2-consecutive u-attracted integ

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A generalized happy function, S e,b maps a positive integer to the sum of its base b digits raised to the e th power. We say that x is a base b, e power, height h, u attracted number if h is the smallest positive integer so that S h e,b (x) = u. Happy numbers are then base 10, 2 power, 1 attracted numbers of any height. Let σ h,e,b (u) denote the smallest height h, u attracted number for a fixed base b and exponent e and let g(e) denote the smallest number so that every integer can be written as x e 1 + x e 2 + ... + x e g(e) for some nonnegative integers x 1 , x 2 , ..., x g(e) . In this paper we prove that if p e,b is the smallest nonnegative integer such that b p e,b > g(e), d = g(e) + 1p e,b , and σ h,e,b (u) ≥ b d , then S e,b (σ h+1,e,b (u)) = σ h,e,b (u).1991 Mathematics Subject Classification. 11A63.