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

Abstract

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