本项目主要研究与斐波那契薛定谔算子和CMV矩阵谱相关的量：谱集的Hausdorff维数、转移指数以及它们之间的关系。具体地，我们研究斐波那契薛定谔算子谱集的Hausdorff维数关于耦合常数的单调性，斐波那契CMV矩阵谱集的局部Hausdorff维数的具体表达式以及它与转移指数之间的关系。斐波那契薛定谔算子是准晶体最主要的数学模型，研究它的谱性质有利于人们了解准晶体中量子波包的传输，而斐波那契CMV矩阵的谱理论可以帮助人们理解一维量子漫步的传播。. 我们知道，在一维算子的谱理论中，尤其是薛定谔算子和CMV矩阵，动力系统中的一些工具起着关键性的作用，如Lyapunov指数，cocycle等。. 本项目的研究对象都是一维的，因此动力系统的方法是我们研究的关键。同时，这两类算子都是与斐波那契序列相关的，我们的研究方法主要依赖于斐波那契迹映射的动力学性质。

This project mainly concerns some quantities related to Fibonacci Hamiltonian and CMV matrices, such as Hausdorff dimension of the spectrum, transport exponents and relations among them. Specifically, we study the monotonicity of Hausdorff dimension of the spectrum of Fibonacci Hamiltonian with respect to the coupling constant, the specific expression of local Hausdorff dimension of the spectrum of Fibonacci CMV matrices and its relation to the transport exponents. As the most prominent model of quasicrystals, the spectral properties of Fibonacci Hamiltonian help us understand the propagation of quantum wavepackets in quasicrystals, while the spectral theory of Fibonacci CMV matrices tells us the spreading rates of one-dimensional quantum walks...As is well known, in the spectral theory of one-dimensional operators, especially Schrodinger operators and CMV matrices, tools in dynamical systems are playing the key role, such as Lyapunov exponents, cocycles etc...The objects in this project are both one-dimensional operators, hence method in dynamical systems is our key in the study. Meanwhile, these two classes of operators are generated by Fibonacci substitution, where our analysis depends heavily on the dynamical properties of the so called Fibonacci trace map.