2022/12/01 by Wu, Sha, Xiao, Yingqing
#28A25 #28A80 (Primary) #42C05 #46C05 (Secondary) #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2212.00340
Given an integer m≥1. Let Σ(m)=\1,2, ⋯, m\ℕ be a symbolic space, and let \(bk,Dk)\k=1m:=\(bk, \0,1,⋯, pk-1\tk) \k=1m be a finite sequence pairs, where integers | bk| , pk≥2, |tk|≥ 1 and pk,t1,t2, ⋯, tm are pairwise coprime integers for all 1≤ k≤ m. In this paper, we show that for any infinite word σ=(σn)n=1∞∈Σ(m), the infinite convolution μσ=δ_b_σ1-1 D_σ1 * δ_(b_σ1 b_σ2)-1 D_σ2 * δ_(b_σ1 b_σ2 b_σ3)-1D_σ3 * ⋯ is a spectral measure if and only if pσn| bσn for all n≥2 and σ∉ \bigcupl=1^∞∏l, where ∏l=\i1i2⋯ ilj∞∈Σ(m): il≠ j, |bj|=pj, |tj|≠1\.