Fourier analysis in chaotic dynamical systems

动力学系统是纯数学和应用数学的一个领域，它分析与时间相关的对象或配置的长期行为，例如细菌种群的进化，流体系统和数字的十进制扩展。如果一个动力系统的初始状态与它的初始状态基本无关，那么这个动力系统就是混沌的。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）的行为来数学形式化。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用方法是使用傅里叶分析将它们分解为更简单的波，并分析它们的频率和幅度。这个想法最近被T. Jordan和T. Sahlsten（监督员）理解一个非常特定的混沌动力系统中的特殊不变分布，称为“高斯映射”。Jordan-Sahlsten特别提出了由高斯映射的特定不变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一类混沌动力系统。具体的第一步将是证明一个类似的结果为“广义高斯映射”，然后继续到更复杂的动力系统的这种。使该项目可行的是最近由J。布尔甘和S。Dyatlov（2017），他将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Bourgain，Dyaltov部分解决了一些技术上的困难所产生的企图概括超越工作的约旦，Sahlsten。

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.