Spectrality of Moran-type measures with staggered contraction ratios

TL;DR

This paper investigates the spectral properties of Moran-type measures with staggered contraction ratios, establishing conditions under which these measures admit orthogonal bases of exponential functions, thus extending existing frameworks.

Contribution

It introduces a new multi-stage decomposition strategy and provides a complete characterization of spectrality for Moran-type measures with specific divisibility and boundedness conditions.

Findings

Proves the measure is unique and generates a Borel probability measure.

Establishes spectrality under divisibility constraints and bounded parameters.

Provides a unified framework extending previous results on spectral measures.

Abstract

Consider a Moran-type iterated function system (IFS) \( \{\phi_{k,d}\}_{d\in D_{2p_k}, k\geq 1} \), where each contraction map is defined as \[ \phi_{k,d}(x) = (-1)^d b_k^{-1}(x + d), \] with integer sequences \( \{b_k\}_{k=1}^\infty \) and \( \{p_k\}_{k=1}^\infty \) satisfying \( b_k \geq 2p_k \geq 2 \), and digit sets \( D_{2p_k} = \{0, 1, \ldots, 2p_k - 1\} \) for all \( k \geq 1 \). We first prove that this IFS uniquely generates a Borel probability measure \( \mu \). Furthermore, under the divisibility constraints \[ p_2 \mid b_2, \quad 2 \mid b_2, \quad \text{and} \quad 2p_k \mid b_k \ \text{for} \ k \geq 3, \] with \(\{b_k\}_{k=1}^\infty\) bounded, we prove that \( \mu \) is a spectral measure, that is, $ L^2(\mu) $ admits an orthogonal basis of exponentials. To fully characterize the spectral properties, we introduce a multi-stage decomposition strategy for spectrums. By imposing the additional hypothesis that all parameters \( p_k \) are even, we establish a complete characterization of \( \mu \)'s spectrality. This result unifies and extends the frameworks proposed in \cite{An-He2014, Deng2022, Wu2024}, providing a generalized criterion for such measures.