自仿测度的谱性、谱对偶性以及非谱性的研究

The present project will consider the theoretical foundation of Fourier analysis on the self-affine sets. It includes the investigation of the spectrality、spectral duality and non-spectrality of self-affine measures. The self-affine measure considered here is the unique invariant probability measure determined by the affine iterated function system with equal weight, and is supported on the self-affine set. The main goal is to characterize the possible conditions for a self-affine measure to be spectral or non-spectral. ..The project has three parts. In Part I, we shall consider the relationship between the spectral pair and compatible pair. Based on the structure of vanishing sum of roots of unity, we want to establish certain relations between the zero set of the Fourier transform of self-affine measure and the existence of compatible pair. We expect to deduce the desired conclusion from such relations and the previously known results. In Part II, we shall deal with the spectral duality relation on the self-affine measures. There are several useful methods to dealing with the spectrality of self-affine measures, such as the method of Ruelle transfer operator、iterated approximation and the decomposition method. We shall apply these methods, combined with the properties of self-affine set which contain some integer points and some cycle points, to answer the spectral duality relation from methodology. Under the condition of non-spectrality of self-affine measure, the Part III is devoted to obtaining a criterion of finiteness or infiniteness on the number of orthogonal exponential functions. We mainly consider the zero set of the symbol function of the digit set, and restrict the discussion within the case that this set is finite in a cube. Such a zero set multiplied by an expanding matrix will play an important role in this part. When an expanding matrix multiplies the zero set many times, the resulting set will become the object of study. The purpose of this part is to know whether or not the resulting set contains an integer point and how many zeros are contained in a given self-affine set. .. The above three research contents will be first studied from the typical fractals, such as the Sierpinski families in the plane and in the sapce. Then we manage to get the desired conclusion from the established result. The research here are fundamental to the spectral or non-spectral problem of self-affine measures. It will lay down the foundation for the theory of Fourier series and Fourier transform established on the fractal sets.

本项目将研究自仿集上建立Fourier分析的理论基础。主要目标是探索支撑在自仿集上自仿测度的谱性、谱对偶关系与非谱性，阐明谱自仿测度产生的条件与性质。研究内容有：1）谱对与和谐对的关系。拟以单位根之和为零的代数结构为基础，建立自仿测度Fourier变换的零点分布特点与和谐对存在性之间的一些联系。2）谱对偶关系。拟从处理谱问题的Ruelle转移算子方法与迭代逼近方法出发，通过矩阵的分解技巧, 结合自仿集上包含整数点与循环点的特征，回答是否存在这种方法上的对偶关系。3）非谱下正交系的有限与无限问题。拟就数字集的符号函数在立方体内的零点是有限的情形，探明扩张矩阵作用下此零点集包含整数点的特征，给出有限与无限的判定条件。上述内容的研究将遵循从特殊到一般的原则，以平面与空间典型分形的研究为基础，给出谱性或非谱性问题一些合理的解答，为分形背景下建立Fourier级数与Fourier变换理论奠定基础。

本项目按照原定的方案与技术路线探讨了自仿测度的谱性、谱对偶关系以及非谱性。主要围绕其中存在的三个猜想，开展了四个方面研究。研究内容与重要结果主要有：（1）研究了自仿测度谱性与和谐对的关系。在Dutkay-Han-Jorgensen猜想的研究方面取得重要进展，指出当谱具有通常给定的形式时，相应的数字集具有谱性，即在一般实数情形下存在和谐对。通过引入广义和谐对，并建立它的一些性质，以及与谱自仿测度的联系，使得人们寻求自仿测度下无限正交指数系的存在性变得更为简单。同时也为用“自仿集上是否包含非零整数点、周期点与循环点”来刻画谱对提供了许多等价的必要条件。（2）研究了和谐对下谱对偶关系的内部结构。解决了困扰大家多年的Dutkay-Jorgensen谱对偶性猜想，获得到了“从一个旧谱得到新谱”的重要方法，并建立了谱之间的一个对偶关系。针对Bernoulli卷积这一典型情形，我们充分利用循环的特点，以及数论中同余关系和有限群中元素阶的概念和性质，获得了这类迭代函数系中循环的特征，给出其乘积谱（也称为尺度谱）的重要刻画，解决了其中的两个公开问题。（3）研究了空间广义Sierpinski垫上自仿测度的谱性或非谱性。获得处理此类问题的一个方法，并探明一些分形集上支撑的自仿测度的谱与非谱性质。解决了空间Sierpinski垫上自仿测度非谱性中有关最大基数的两个问题，完整的刻画了其上自仿测度的非谱性。(4)研究了非谱测度下正交指数函数系的有限性与无限性的条件。给出空间自仿测度下存在无限正交指数系的一个刻画，得到有限正交指数函数系的一个充分条件。对数字集的符号函数的零点集合在扩张矩阵作用下的变化规律有了比较好的掌握。在实数情形下，这些集合之间若存在一个包含关系，那么这类自仿测度是非谱的，并可知非谱的类型。进一步，通过对此类集合特征分析以及非零中间点性质的应用，得到了非谱测度下正交指数系基数的一个更为精确的估计。以上研究内容丰富和发展了自仿测度的谱理论，为自仿集上建立Fourier分析理论奠定了坚实的基础。