Spectral properties of a class of Moran measures

Abstract

Let B = { b n } n ∈ N and Q = { q n } n ∈ N be a sequence of positive integers respectively satisfying b n ≥ q n > 1 . Let μ B , Q = : δ b 1 − 1 { 0 , 1 , … , q 1 − 1 } ⁎ δ ( b 1 b 2 ) − 1 { 0 , 1 , … , q 2 − 1 } ⁎ ⋯ . μ B , Q is called a Moran measure , which is a generalization of self-affine measures. In particular, we call μ B , Q the N-Bernoulli convolution if b n = b and q n = N with 1 < b ∈ R for all n ∈ N . It is known [1] that μ B , Q is a spectral measure if q n divides b n for all n ∈ N . In this paper, we give a simpler proof to this result by developing the method given by Dai [3] and simultaneously, we construct a spectrum for the measure μ B , Q when q n divides b n for all n ∈ N . Moreover, we give an equivalent condition for the N -Bernoulli convolution with 1 < b ∈ R and a sufficient condition for the Moran-type measures μ B , Q with 1 < b n ∈ Z to admit an infinitely many orthogonal exponentials. We also provide an equivalent characterization for the maximal orthogonal exponential sets of μ B , Q by defining a mixed tree mapping τ of the μ B , Q , which is an important tool for us to characterize the maximality of orthogonal exponential set.