, ) : 1 or p, that each operator a3'i is a Schrôdinger operator with substitutional potential. Nforeover, when p : l-, lhe corresponding substitution is again {. and the nature of the spectrum of these /sparse Schrôdinger operators is the same as in the classical case of one-dirnensional discrete Schrôdinger operators with the same substitutional potential, !'e first consider the case ô(p) = 1. By (7.6) and Proposition 7.3, we have to study a new family of Schrôdirrger operators (H3'0).e xtel, associated with the strictly ;r,-ergodic rlynamical system Tp

The period-rloubling stt'bst'ituti'on We consider the alphabsl ,4: {0.1}. The period doubling substitution is definecl by, pp.0-01 ,

, It is a primitive substitution with coustant length equal to 2. \Me can choose the fixed-poinr

, By primitivity of {, f (i) is generated by any fixed-point, and the dynamical system generated by { is strictly ergodic. Moreover. the decomposition function can be calculated (for more details see lzl), 6(2") :2" Vn 2 I 6(m) :1 , if m is odd 6(2"m):2n, Vn)1, Vmodd

Q: (ii) /br arLA n,on,-negatiue integer n. I2'' i,s a Cantor set of zero Lebesgue measttre. art,d for p-alrnost all r in f. il is purely si,ngular con,tinuous.Proof. (i) is a direct consequence of Theorem 7.4. For (ii), we remark that each Ilf'' is a Schrôdinger operator with substitutional potential generated by the period doubling substitution, Let us cons'ider the peri,od dou,bling subst,'itrtt'ion on A : {0,1}. Then (i) for anlq (r oPerat,or r (1i) for anu n, sure. att'd (i) for any (non-negatiue) odd 'integer p. \)e is the spectrum ,

, By primitivity of {,2(i) is generated by any fixed-point, and the dynarnical systenr generated by ( is strictly ergodic. Moreover. the decomposition function can be calculated: 6(2"1 :2, p.1

, According to Theorem 7.4, we find an analogue of Proposition 7.5: Proposition 7.6. Let us cons'ider th,e Thue Morse substitut'ion on A : {0. 1}. Then (i) for any (non-negati,ue) od,d integer p, \p i,s the spectrum of a Schrôd, 1 , if nz is odd 6(2"m):2n, Vn>1, Vmodd, vol.6

, (ri) for a,ng non-negatiue integer rn. \2"' is a Cantor set, of zero Lebesgue rneasure, and for p-almost all r in X

, Conclusion We can now conlpare the three special cases of ?Èsparse Schrôdinger operators treated in this pàper. When the potential is a sequence of independent identically distributed random variables, neither the nature of the spectrum nor its Iocation change u'ith p: the spectrum of f

,

, On the other side, if the potential is l/-periodic. then the nature of the spectrurn does not change with p

, In fact we only know that there exists no absolutely continuous part in the spectrum of Ht' rf 6(p) = 1 or p. Moreover. if the primitive substitution has a constatrt length I and if p-0". then the nature of !p is the one of the spectrum of the corresponding substitutional Schrôdinger operator fI. Finally, in these cases (random, periodic and substitutional), the spectral behavior of trrsparse Schrôdinger operators is similar to the one of the corresponding Schrôdinger operators for all p when the potential is random or periodic, and for 6(p) = 1 or p when it is substitutional. We could conjecture similar results for sparse Schrôdinger operators vl'ith limit periodic or quasi-periodic potentials, mod (l/) or p = À"-1 mod (À ), p.1

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