Fourier analysis in chaotic dynamical systems

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.