| Abstrakt: |
Let μ { R i } , { D i } be the probability measure generated by the iterated function system (IFS): { F R i , D i (x) = R i − 1 (x + d) : d ∈ D i } i = 1 ∞ , where R i = ρ ( 1 d i 0 1 ) is an expanding matrix with 1 < ρ ∈ ℝ, di ∈ ℤ, and D i = { ( 0 0 ) , ( 1 0 ) , ( k i 1 ) } with ki ∈ ℤ, and supi∈ℕ{∣di∣, ∣ki∣} < ∞. In this paper, we consider the spectral properties of μ { R i } , { D i } , we show that μ { R i } , { D i } is a spectral measure, i.e., there exists a countable set Λ ⊆ ℝ2, such that E(Λ) ≔ {e2πi〈x,λ〉, λ ∈ Λ} forms an orthonormal basis for L 2 (μ { R i } , { D i } ) , if and only if ρ = 3k for some k ∈ ℕ. Furthermore, we also provide an equivalent characterization for the maximal bi-zero set for μ { R i } , { D i } by defining a mixed tree mapping for it. And we also obtain some results associated with the Sierpinski-type measures. [ABSTRACT FROM AUTHOR] |
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