Spectral properties of a class of Moran measures on [formula omitted

Abstract

Given a pair (R,D), where R={Ri}i=1∞ is a sequence of expanding matrix (i.e., all the eigenvalues of Ri have modulus strictly greater than 1), and D={Di}i=1∞⊆Z2. It is well known that there exists an infinite convolution generated by (R,D) which satisfies μR,D≔δR1−1D1∗δ(R2R1)−1D2∗⋯,we say that μR,D is a Moran measure if it convergent to a probability measure with compact support in a weak sense, where δE=1#E∑d∈Eδd is the uniformly discrete measure on E. In this paper, we consider the spectral properties of the Moran measure μR,D with R=b100b2, and #Di=p, where 1<b1,b2∈R and p is a prime number, supi∈N,d∈Di|d|<∞. Let B(p):≡{1pjk:1≤j,k≤p−1} and Z(δˆE)={x∈R2:δˆE(x)=0}. We show that under the conditions that Z(δˆDi)=⋃k=1φ(i)Zk(p)for some φ(i)∈N, where Zk(p)⊆B(p)+Z2 with (Zk(p)−Zk(p))∖Z2⊆Zk(p), and Zk(p)⁄⊆B(q)+Z2 for 0<q<p, 1≤k≤φ(i), {x∈[0,1)2:|δˆDi(x)|=1}={00}. Then μR,D is a spectral measure if and only if R=pk100pk2 for some k1,k2∈N. This extends the results of Sierpinski-type measure considered by Dai et.al [ACHA,2021]. Also some sufficient conditions for μR,D being a spectral measure when p is not a prime number are given.