On the Fourier Orthonormal Bases of a Class of Moran Measures on ℝ²

Abstract

Let ${\mu _{\left\{{{R_i}} \right\},\left\{{{{\cal D}_i}} \right\}}}$ be the probability measure generated by the iterated function system (IFS): $\left\{{{F_{{R_i},{{\cal D}_i}}}\left(x \right) = R_i^{- 1}\left({x + d} \right):d \in {{\cal D}_i}} \right\}_{i = 1}^\infty$, where ${R_i} = \rho \left({\matrix{1 & {{d_i}} \cr 0 & 1 \cr}} \right)$ is an expanding matrix with 1 < ρ ∈ ℝ, d_i ∈ ℤ, and ${{\cal D}_i} = \left\{{\left({\matrix{0 \cr 0 \cr}} \right),\left({\matrix{1 \cr 0 \cr}} \right),\left({\matrix{{{k_i}} \cr 1 \cr}} \right)} \right\}$ with k_i ∈ ℤ, and sup_i∈ℕ{∣d_i∣, ∣k_i∣} < ∞. In this paper, we consider the spectral properties of ${\mu _{\left\{{{R_i}} \right\},\left\{{{{\cal D}_i}} \right\}}}$, we show that ${\mu _{\left\{{{R_i}} \right\},\left\{{{{\cal D}_i}} \right\}}}$ is a spectral measure, i.e., there exists a countable set Λ ⊆ ℝ², such that E(Λ) ≔ {e^{2πi〈x,λ〉}, λ ∈ Λ} forms an orthonormal basis for ${L^2}\left({{\mu _{\left\{{{R_i}} \right\},\left\{{{{\cal D}_i}} \right\}}}} \right)$, if and only if ρ = 3k for some k ∈ ℕ. Furthermore, we also provide an equivalent characterization for the maximal bi-zero set for ${\mu _{\left\{{{R_i}} \right\},\left\{{{{\cal D}_i}} \right\}}}$ by defining a mixed tree mapping for it. And we also obtain some results associated with the Sierpinski-type measures.