In 2017, BourgainDyatlov [8] prove that PattersonSullivan 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 BourgainDyatlov which stated that a technical dimension assumption can be removed from a Fourier decay theorem of JordanSahlsten [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 FrÂ´edÂ´eric Naud. All aforementioned work is joint with Tuomas Sahlsten. To finish, we go back to the groundbreaking work of BourgainDyatlov, 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 JordanSahlsten 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 BourgainDyatlov. 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 BourgainDyatlov [8].
Date of Award  1 Aug 2022 

Original language  English 

Awarding Institution   The University of Manchester


Supervisor  Tuomas Sahlsten (Supervisor) & Charles Walkden (Supervisor) 

 Schottky Structure
 Salem Sets
 Round Sets
 Ergodic Theory
 Thermodynamical Formalism
 Additive Combinatorics
 Fractal
 Real Analysis
 SumProduct Theory
 PattersonSullivan Measure
 BowenSeries Map
 SelfConformal
 Badly Approximable Numbers
 Dirichlet's Theorem
 Fractal Uncertainty
 Total Nonlinearity
 NonConcentrated Derivative
 Exponential Sum
 Fourier Transform
 Fourier Decay
 Nonlinear Dynamics
 Dynamical Systems
 Measure
 Large Deviations
 Gauss map
 Fourier Dimension
 Dimension
 Fuchsian Group
 Hyperbolic Geometry
 Gibbs Measure
 Transfer Operator
 Complex Transfer Operator
 Diophantine Approximation
 Multiplicative Convolution
Fourier Decay in Nonlinear Dynamics
Stevens, C. (Author). 1 Aug 2022
Student thesis: Phd