2023/05/20 by Jinsong Liu, Liu, Jinsong, Zheng-Yi Lu +3
Mathematics · Materials Science · #Mathematical Dynamics and Fractals #Analytic and geometric function theory #Quasicrystal Structures and Properties
paper · pdf · doi:10.48550/arxiv.2305.12135
Let \(pn, Dn, Ln)\ be a sequence of Hadamard triples on ℝ. Suppose that the associated Cantor-Moran measure μ_\pn,Dn\=δp1-1D1∗δ(p2p1)-1D2∗⋯, where supn\|pn-1d|:d∈ Dn\<∞ and sup#Dn<∞. It has been observed that the spectrality of μ_\pn,Dn\ is determined by equi-positivity. A significant problem is what kind of Moran measures can satisfy this property. In this paper, we introduce the conception of Double Points Condition Set (DPCS) to characterize the equi-positivity equivalently. As applications of our characterization, we show that all singularly continuous Cantor-Moran measures are spectral. For the absolutely continuous case, we study Fuglede's Conjecture on Cantor-Moran set. We show that the equi-positivity of μ_\pn,Dn\ implies the tiling of its support, and the reverse direction holds under certain conditions.