Common Spectral Eigenvalue Spectra for Random Convolutions Generated by Hadamard Triples

Abstract

Lu proved that the set $$\mathcal{T}:=\{t\in \mathbb{Z}\setminus \{0\}:(q,\mathcal{D},tL)\text{ forms a Hadamard triple}\}$$ constitutes a spectral eigenvalue set for $\hat{I}{¼}_{q,D}$, where $(q,\mathcal{D},L)$ is a Hadamard triple. And they prove for $s\in [0,\frac{\log \#\mathcal{D}}{\log q}]$, the corresponding spectra form a family of cardinality continuum. In this paper, we study Moran measures formed by random convolutions of finite Hadamard triples. Let $$\hat{I}¼=\hat{I}{´}_{M_1^{-1}{D}_1}\ast \hat{I}{´}_{M_2^{-1}{D}_2}\ast \cdots ,M_k=q_1{q}_2\cdots q_k,$$ where the factors are produced from finitely many triples $\{(N_j,B_j,L_j):1\le j\le m\}$, $(\ddot{I}{‰}_k{)}_{k=1}^{\infty }\in \{1,2,\dots ,m{\}}^{\mathbb{N}}$, and $n_k\in {\mathbb{N}}^{+}$, by setting $$q_k=N_{\ddot{I}{‰}_k}^{n_k},D_k=B_{\ddot{I}{‰}_k},E_k=N_{\ddot{I}{‰}_k}^{n_k-1}{L}_{\ddot{I}{‰}_k}.$$ Assume a non-full-digit gap $$\ddot{I}:=\underset{1\le j\le m}{\min }\frac{N_j}{\#B_j}>1.$$ For the common Hadamard triple multiplier set $${\mathcal{T}}_{\ast }:=\bigcap _{j=1}^m\{t\in \mathbb{Z}\setminus \{0\}:(N_j,B_j,t{L}_j)\text{ is a Hadamard triple}\},$$ Our main result is that, for every $$0\le s\le \hat{I}{º}_{\ddot{I}}‰:=\underset{R\to \infty }{\mathrm{limsup}}\frac{{\sum }_{r=1}^R\log \#D_r}{{\sum }_{r=1}^R\log q_r},$$ there exist continuum many countable sets $\hat{I}›\subset \mathbb{Z}$ such that $t\hat{I}›$ is a spectrum of $\hat{I}¼$ for every $t\in {\mathcal{T}}_{\ast }$ and ${\dim }_{Be}\hat{I}›=s$.