论文标题
$ \ mathbb {r} $上的一类无限卷积的光谱
Spectrality of a class of infinite convolutions on $\mathbb{R}$
论文作者
论文摘要
给定一个整数$ m \ geq1 $。令$σ^{（m）} = \ {1,2，\ cdots，m \}^{\ mathbb {n}} $是一个符号空间，让$ \ {（b_ {k}，d_ {k}，d_ {k}） \ {0,1，\ cdots，p_ {k} -1 \} t_ {k}）\} _ {k = 1}^{m} $是有限序列对，其中整数是$ | | B_ {K} | $，$ p_ {k} \ geq2 $，$ | t_ {k} | \ geq 1 $和$ p_ {k {k}，t_ {1}，t_ {2}，\ cdots，t_ {m} $是所有$ 1 \ leq k k \ leq leq m $。在本文中，我们表明，对于任何无限单词$σ= \ left（σ_{n} \右） d_ {σ_{1}}} *Δ b_ {σ_{2}} b_ {σ_{σ_{3}} \ right）^{ - 1} d_ {σ_{σ_{3}}} * \ cdots $$是光谱度量，并且仅当$ p_ {σ_n} \ bigCup_ {l = 1}^\ infty \ infty \ prod_ {l} $，其中$ \ prod_ {l} = \ {i_ {1} i_ {2} \ cdots i_ {l} | b_ {j} | = p_ {j}，| t_ {j} | \ neq1 \} $。
Given an integer $m\geq1$. Let $Σ^{(m)}=\{1,2, \cdots, m\}^{\mathbb{N}}$ be a symbolic space, and let $\{(b_{k},D_{k})\}_{k=1}^{m}:=\{(b_{k}, \{0,1,\cdots, p_{k}-1\}t_{k}) \}_{k=1}^{m}$ be a finite sequence pairs, where integers $| b_{k}| $, $p_{k}\geq2$, $|t_{k}|\geq 1$ and $ p_{k},t_{1},t_{2}, \cdots, t_{m}$ are pairwise coprime integers for all $1\leq k\leq m$. In this paper, we show that for any infinite word $σ=\left(σ_{n}\right)_{n=1}^{\infty}\inΣ^{(m)}$, the infinite convolution $$ μ_σ=δ_{b_{σ_{1}}^{-1} D_{σ_{1}}} * δ_{\left(b_{σ_{1}} b_{σ_{2}}\right)^{-1} D_{σ_{2}}} * δ_{\left(b_{σ_{1}} b_{σ_{2}} b_{σ_{3}}\right)^{-1}D_{σ_{3}}} * \cdots $$ is a spectral measure if and only if $p_{σ_n}\mid b_{σ_n}$ for all $n\geq2$ and $σ\notin \bigcup_{l=1}^\infty\prod_{l}$, where $\prod_{l}=\{i_{1}i_{2}\cdots i_{l}j^{\infty}\inΣ^{(m)}: i_{l}\neq j, |b_{j}|=p_{j}, |t_{j}|\neq1\}$.
