Fourier Decay in Nonlinear Dynamics

Abstract

In 2017, Bourgain-Dyatlov [8] prove that Patterson-Sullivan measures on the limit set of convex cocompact Fuchsian groups have polynomial Fourier decay. We begin by proving that their main tool, Bourgain's exponential sum theory, can be used to prove polynomial Fourier decay for Gibbs measures for sufficiently nonlinear Markov maps. We follow up by proving a remark of Bourgain-Dyatlov which stated that a technical dimension assumption can be removed from a Fourier decay theorem of Jordan-Sahlsten [26] by proving that the Gauss map is sufficiently nonlinear. We move on to prove an analogous theorem for a much more general class of (finite) nonlinear Markov maps with a strong separation condition. We do so using the complex transfer operator theory of Naud [44] as recommended to us by Jialun Li and Fr´ed´eric Naud. All aforementioned work is joint with Tuomas Sahlsten. To finish, we go back to the ground-breaking work of Bourgain-Dyatlov, and ask whether we can prove their main Fourier decay result for Gibbs measures on limit sets of convex cocompact Fuchsian groups. A corollary of the aforementioned theorem on general nonlinear Markov maps with strong separation is that we can obtain polynomial Fourier decay for such measures. Alternatively, we can use the combinatorial large deviation theory of Jordan-Sahlsten to prove a polynomial Fourier decay theorem for a class of measures which are defined using Schottky structures. This avoids the need for complex transfer operator theory to prove that the Fuchsian groups are sufficiently nonlinear; we can just use the distortion factor analysis of Bourgain-Dyatlov. We conclude by proving a fractal uncertainty principle for Gibbs measures for Markov maps with (eventual) polynomial Fourier decay by slightly adapting a proof of Bourgain-Dyatlov [8].

Date of Award	1 Aug 2022
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Tuomas Sahlsten (Supervisor) & Charles Walkden (Supervisor)