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

Gaétan Leclerc

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.

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.

Paper Structure

This paper contains 84 sections, 199 theorems, 1320 equations.

Table of Contents

  1. Introduction
  2. Fractal geometry, hyperbolic dynamics, and harmonic analysis
  3. Hausdorff dimension and Fourier dimension
  4. The upper and lower Fourier dimension of a measure
  5. A variation with real valued phases
  6. A directional variation
  7. What about sets ?
  8. Some sets with positive Fourier dimension.
  9. Consequences of positivity for the Fourier dimension
  10. The Thermodynamical formalism
  11. Equilibrium states on shift spaces
  12. Suspension flows and Dolgopyat's estimates
  13. Additive combinatorics: the sum-product phenomenon
  14. Additive combinatorics in $\mathbf{F}_p$
  15. The sum-product phenomenon and fractal measures
  16. ...and 69 more sections

Key Result

Theorem 1.1.5

Let $X \subset \mathbb{R}^d$ be a compact set. Denote by $\mathcal{P}(X)$ the set of all borel probability measures supported on $X$. Then:

Theorems & Definitions (488)

  • Definition 1.1.1: Hausdorff measure
  • Remark 1.1.2
  • Definition 1.1.3: Hausdorff dimension
  • Remark 1.1.4
  • Theorem 1.1.5: Frostman's lemma
  • Remark 1.1.6
  • Definition 1.1.7
  • Theorem 1.1.8: Frostman's lemma, revisited
  • Remark 1.1.9
  • Definition 1.1.10: Fourier dimension
  • ...and 478 more