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

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.