Spectral number of 3-Bernoulli convolutions on R and Sierpinski-type measures on R2

摘要

Let mu(R,D) be a class of Sierpinski-type measures generated by a pair (R, D), where R =((b1)(0) (b2)(0)) with b(i) is an element of R and b(i) > 1, i = 1, 2 and D = {((0)(0)), ((0)(1)), ((1)(0))}. And let mu(b) be the 3-Bernoulli convolutions on R determined by the pair (b,{0, 1, 2}) with 1 < b is an element of R. It has been shown that mu(R,D) and mu(b) admit an infinitely many exponential mutually orthogonal system if and only if b(i) = r(i)root 3k(i)/q(i) with gcd(q(i), 3k(i)) = 1, i = 1, 2 and b = r root 3k/q with gcd(3k, q) = 1 respectively. In this paper, we will study the maximal number of exponentials of orthogonal sets of L-2(mu(R,D)) and L-2(mu(b)) which we call the spectral number of mu(R,D) or mu(b). In view of the connection of orthogonality between Sierpinski-type measures mu(R,D) on R-2 and 3-Bernoulli convolutions mu(b) on R, we study the spectral number of mu(b) according to the cut-off point b = r root 3k/q. Based on the results for mu(b), we give a classification on the spectral number of all Sierpinski-type measures mu(R,D) except for the case that at least one b(i) is not an element of Q and it is not in the form of r(i)root p(i)/q(i) with gcd(p(i), q(i)) = 1. In addition, we provide a structure theorem on the exponential orthogonal sets in L-2(mu(R,D)) for b(i) = r(i)root 3k(i)/q(i), i = 1, 2 and at least one b(i) = r(i)root p(i)/3k(i) and that in L-2( mu(b)) for b = r root 3k/q and b = r root p/3k. To the end, we give an explicit representation on the maximal orthogonal set of exponentials for a class of Moran measures mu(w) by defining a mixed tree map over a symbol space. As an application, all maximal orthogonal sets of exponentials of mu(R,D) with the rational R = ((3k1/q1)(0) (0)(3k2/q2)) can be explicitly expressed. This result improves the characterization of maximal orthogonal set of exponentials for the integral matrix R = ((3k1)(0) (0)(3k2)) to that for the rational matrix R = ((3k1/q1)(0) (0)(3k2/q2)).

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