Spectral properties of certain Moran measures with consecutive and collinear digit sets

Let the {2\times 2} expanding matrix {R_{k}} be an integer Jordan matrix, i.e., {R_{k}=\operatorname{diag}(r_{k},s_{k})} or {R_{k}=J(p_{k})}, and let {D_{k}=\{0,1,\ldots,q_{k}-1\}v} with {v=(1,1)^{T}} and {2\leq q_{k}\leq p_{k},r_{k},s_{k}} for each natural number k. We show that the sequence of Hadamard triples {\{(R_{k},D_{k},C_{k})\}} admits a spectrum of the associated Moran measure provided that {\liminf_{k\to\infty}2q_{k}\lVert R_{k}^{-1}\rVert<1}.

A CLASS OF SPECTRAL MORAN MEASURES ON ℝ

Let [Formula: see text] be an arithmetic digit set for each [Formula: see text], where [Formula: see text], and let [Formula: see text] be a sequence of integers larger than 1. In this paper, we prove that the Moran measure [Formula: see text] generated by infinite convolution of finite atomic measures [Formula: see text] is a spectral measure if [Formula: see text] and [Formula: see text].

Spectrality of a Class of Moran Measures

Let $\{M_{n}\}_{n=1}^{\infty }$ be a sequence of expanding matrices with $M_{n}=\operatorname{diag}(p_{n},q_{n})$, and let $\{{\mathcal{D}}_{n}\}_{n=1}^{\infty }$ be a sequence of digit sets with ${\mathcal{D}}_{n}=\{(0,0)^{t},(a_{n},0)^{t},(0,b_{n})^{t},\pm (a_{n},b_{n})^{t}\}$, where $p_{n}$, $q_{n}$, $a_{n}$ and $b_{n}$ are positive integers for all $n\geqslant 1$. If $\sup _{n\geqslant 1}\{\frac{a_{n}}{p_{n}},\frac{b_{n}}{q_{n}}\}<\infty$, then the infinite convolution $\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}}=\unicode[STIX]{x1D6FF}_{M_{1}^{-1}{\mathcal{D}}_{1}}\ast \unicode[STIX]{x1D6FF}_{(M_{1}M_{2})^{-1}{\mathcal{D}}_{2}}\ast \cdots \,$ is a Borel probability measure (Cantor–Dust–Moran measure). In this paper, we investigate whenever there exists a discrete set $\unicode[STIX]{x1D6EC}$ such that $\{e^{2\unicode[STIX]{x1D70B}i\langle \unicode[STIX]{x1D706},x\rangle }:\unicode[STIX]{x1D706}\in \unicode[STIX]{x1D6EC}\}$ is an orthonormal basis for $L^{2}(\unicode[STIX]{x1D707}_{\{M_{n}\},\{{\mathcal{D}}_{n}\}})$.

Spectrality of Moran measures with finite arithmetic digit sets

Let [Formula: see text] be a prime and [Formula: see text] be a sequence of finite arithmetic digit sets in [Formula: see text] with [Formula: see text] uniformly bounded, and let [Formula: see text] be the discrete probability measure on the finite set [Formula: see text] with equal distribution. For [Formula: see text], the infinite Bernoulli convolution [Formula: see text] converges weakly to a Borel probability measure (Moran measure). In this paper, we study the existence of exponential orthonormal basis for [Formula: see text].

EXPONENTIAL ORTHONORMAL BASES OF CANTOR–MORAN MEASURES

Let [Formula: see text] be a Cantor–Moran measure given by the infinite convolution of finite measures with equal probability [Formula: see text] where [Formula: see text] and [Formula: see text] for [Formula: see text] In this paper, we present a complete characterization for maximal orthogonal sets of exponentials of [Formula: see text] in terms of maximal mappings. As its application, we give a sufficient condition for a maximal orthogonal set to be a basis.

THE MULTIFRACTAL DIMENSION FUNCTIONS OF HOMOGENEOUS MORAN MEASURE

In this paper we attain the multifractal dimension functions of Moran measure associated with homogeneous Moran fractals and give a sufficient condition which ensures the multifractal spectrum of Moran measure is equal to the Legendre transform of the multifractal dimension functions. As an application of this method, we give an example and a counterexample about the validity of the multifractal formalism of Moran measure.