Supervisor: Prof Oliver Jenkinson
Project description:
This project is rooted in Ergodic Theory and Dynamical Systems. The mathematics of Ergodic Theory originally emerged from 19th century physicists' desire to equate time averages with space averages (Boltzmann's ergodic hypothesis), and the influence of Ergodic Theory within Dynamical Systems is increasingly pronounced. This project seeks to make progress in a topical and active branch of Ergodic Theory and Dynamical Systems known as Ergodic Optimization, which aims to investigate those orbits, and those invariant probability measures, that optimize time averages. For example, given a dynamical system (a map or flow) on a space, and a given subset A of that space, we will be interested in determining which orbits spend most time in A. More generally, if f is a real-valued function defined on the space, we will seek to determine which orbits, or which invariant probability measures, lead to the largest f-average. Computer experiment has played a major role in stimulating research in this area, but in recent years there has been exciting progress in understanding the theoretical structure of such optimizing orbits and measures.
In this project there will be scope for both theoretical and experimental research, with the balance between these approaches being determined by the student's preferences.
At an experimental level, the student will be involved in computer experiments aimed at generating data on optimizing orbits for specific dynamical systems, and specific functions f. At a theoretical level, a major conjecture about the importance of periodic orbits in Ergodic Optimization was solved around 8 years ago (Gonzalo Contreras, `Ground states are generically a periodic orbit', Inventiones Mathematicae, 2016). More recently, the Chinese team of Wen Huang, Zeng Lian, Xiao Ma, Leiye Xu, and Yiwei Zhang have pioneered a new approach, and succeeded in showing that periodic orbits play a key role as optimizing orbits for a wide range of dynamical systems enjoying `uniform hyperbolicity'. An orienting conjecture, formulated by this PhD project's supervisor Oliver Jenkinson, is the Typically Periodic Optimizing Conjecture (or TPO Conjecture). Work in progress by the supervisor, together with collaborators in China, is currently exploring the extent to which the TPO phenomenon is true for more general dynamical systems.
The student will be involved in working, both individually and as a team member, on theoretical aspects of the TPO Conjecture for a range of dynamical systems which enjoy some kind of hyperbolic or chaotic properties.
Further information: How to apply Entry requirements Fees and funding