三元数字集的自相似测度的谱性性质

Spectrality of Self-Similar Measures with Three Element Digit Sets

摘要: Fu和Wen证明了压缩比为实数ρ和有界三元整数字集列D_n={0, a_n,b_n}⊂ℤ生成的无穷Bernoulli 卷积测度是谱测度的充要条件.本文研究由压缩比为实数ρ和三元实数字集D定义的迭代函数系统生成的自相似测度的谱性质, 我们证明该测度是谱测度当且仅当ρ⁻¹是以3为因子的非零整数且存在非零实数a, 使得a(D−α)模3同余集合{0,1,2}, 其中α∈D 。

Abstract: Fu and Wen prove that the convolution of the infinite Bernoulli measure generated by the compression ratio of real numbers ρ and the sequence of bounded three-element integers D_n={0, a_n,b_n}⊂ℤ is a sufficient and necessary condition for spectral measure. In this paper we study the spectrality of the self-similar measure generated by the iterative function system defined by the compression ratio of real numbers ρ and the set of three-element real digits D . We prove that the measure is spectral if and only if ρ⁻¹ is a non-zero integer with a factor of  3 and a(D−α) is congruence with {0,1,2} under (mod 3) for some a, where α∈D.