Fourier analysis in chaotic dynamical systems

动力系统是纯数学和应用数学中的一个领域，它分析依赖时间的对象或构型的长期行为，如细菌种群、流体系统的演化和数字的十进制展开。如果动力系统基本上独立于系统的初始状态，则称其为混沌。混沌动力学可以通过研究动力学系统的不变分布(初始状态的分布不随时间变化的分布)的行为而在数学上形式化。这是从物理中经典力学的哈密顿公式得到启发的。混沌动力系统的不变分布可以表现出非常复杂的行为，因此已经开发了各种数学工具来理解它们的精细结构。在信号处理中，研究复杂信号的一种常见方法是使用傅立叶分析将它们分解成更简单的波，并分析它们的频率和幅度。这一想法最近被T·乔丹和T·萨尔斯滕(主管)用来理解一个非常具体的混沌动力系统中的特殊不变分布，称为“高斯映射”。特别地，Jordan-Sahlsten建立了由Gauss映射的特定不变分布构造的波在快速振荡时肯定具有很小的振幅。本博士项目试图将Jordan-Sahlsten的工作进一步发展到更广泛的一类混沌动力系统。具体的第一步将是证明“广义高斯映射”的类似结果，然后继续研究这类更复杂的动力系统。使该项目可行的是J.Bourain和S.Dyatlov(2017)最近的开创性工作，他们将Jordan-Sahlsten的工作与量子力学中的一个类似问题联系起来。特别是，布尔加恩-达阿尔托夫部分解决了一些技术困难，这些困难是由于试图在乔丹-萨尔斯滕的工作之外进行概括而产生的。

Dynamical systems is a field in pure and applied mathematics that analyse the long-term behaviour of time-dependent objects or configurations such as the evolution of bacteria populations, fluid systems and the decimal expansions of numbers. The dynamical system is called chaotic if it is mostly independent of the initial state of the system. Chaotic dynamics can be mathematically formalised by studying the behaviour of invariant distributions for the dynamical system (distributions of initial states that do not change in time). This is inspired by the Hamiltonian formulation of classical mechanics in physics.Invariant distributions for chaotic dynamical systems can exhibit very complicated behaviour and thus various mathematical tools have been developed to understand their fine structure. In signal processing one common way to study complicated signals is to use Fourier analysis to decompose them into simpler waves and analyse their frequencies and amplitudes. This idea was recently applied by T. Jordan and T. Sahlsten (the supervisor) to understand special invariant distributions in a very specific chaotic dynamical system called the "Gauss map". Jordan-Sahlsten in particular established that the waves constructed from the specific invariant distribution for Gauss map must have very small amplitudes when the wave is oscillating rapidly.This PhD project attempts to develop further the work of Jordan-Sahlsten to a much wider class of chaotic dynamical systems. The concrete first step would be to prove an analogous result for a "generalised Gauss map" and then continue to more complicated dynamical systems of this kind. What makes the project feasible is the recent groundbreaking work by J. Bourgain and S. Dyatlov (2017), who connected the work of Jordan-Sahlsten to an analogous problem in quantum mechanics. In particular Bourgain-Dyaltov addressed partially some of the the technical difficulties arising from the attempts to generalise beyond the work of Jordan-Sahlsten.