Fourier analysis in chaotic dynamical systems

动力系统是纯数学和应用数学的一个领域，用于分析依赖时间的物体或结构的长期行为，如细菌种群的进化、流体系统和数字的十进制展开。如果动力系统基本上与系统的初始状态无关，则称为混沌系统。混沌动力学可以通过研究动力系统的不变分布（不随时间变化的初始状态分布）的行为在数学上形式化。这是受到物理学中经典力学的哈密顿公式的启发。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用方法是使用傅里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。这个想法最近被T. Jordan和T. Sahlsten（导师）应用于理解一个非常特殊的混沌动力系统中的特殊不变量分布，称为“高斯映射”。特别是Jordan-Sahlsten建立了由高斯图的特定不变量分布构造的波在快速振荡时必须具有非常小的振幅。这个博士项目试图将Jordan-Sahlsten的工作进一步发展到更广泛的混沌动力系统。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. Bourgain和S. Dyatlov（2017）最近的突破性工作，他们将Jordan-Sahlsten的工作与量子力学中的类似问题联系起来。特别是，Bourgain-Dyaltov部分地解决了由于试图在Jordan-Sahlsten的工作之外进行概括而产生的一些技术困难。

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.