Abstract
For an expanding real matrix M=[ρ−1C0ρ−1]∈M2(R) and a digit set D={(0,0)t,(1,0)t,(0,1)t}, let μM,D be the self-affine measure generated by M and D. In this paper, we show that L2(μM,D) admits an infinite orthogonal set of exponential functions if and only if |ρ|=(q/p)1r and C=κρ−1 for some positive integers p,q,r with p∈3Z, gcd(p,q)=1 and κ∈Q. Moreover, if L2(μM,D) does not admit any infinite orthogonal set of exponential functions, we estimate the number of orthogonal exponential functions in L2(μM,D) and give the exact maximal cardinality.
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