Nonlinearity, Fractals, Fourier decay -- Harmonic analysis of equilibrium states for hyperbolic dynamical systems

TL;DR

This PhD manuscript reviews and extends harmonic analysis techniques applied to equilibrium states in hyperbolic dynamical systems, including new proofs of Fourier dimension positivity for nonlinear surface diffeomorphisms.

Contribution

It provides the first proof of Fourier dimension positivity for basic sets of nonlinear, area-preserving, smooth Axiom A surface diffeomorphisms, extending previous work with new methods.

Findings

First proof of Fourier dimension positivity for nonlinear Axiom A diffeomorphisms

Generalizations of previous arXiv papers on harmonic analysis and dynamical systems

New results on Fourier decay and fractal geometry in hyperbolic dynamics

Abstract

This is my (reviewed) PhD manuscript. It contains 6 Chapters, which contains mostly already published work, except for Chapter 5 which is new. Chapter 1 introduce basic notions on fractal geometry: the Fourier dimension, the thermodynamical formalism and additive combinatorics. Chapter 2 is a generalized version of arXiv:2211.08088. Chapter 3 is a slightly upgraded version of arXiv:2112.00701. Chapter 4 is a generalized version of arXiv:2301.10623. Chapter 5 and Chapter 4 together contains the first proof of the positivity of the Fourier dimension for basic sets of nonlinear, area-preserving, smooth Axiom A diffeomorphisms on surfaces. This is possible by adapting some ideas that can be found in Tsujii-Zhang's work arXiv:2006.04293. Some future work still needs to be done to prove that the nonlinearity conditions are generic in this setting: this should become a proper research article in the future. Chapter 6 is a slighty upgraded version of arXiv:2307.10755.