Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral gaps and Fourier decay for self-conformal measures in the plane

Amir Algom, Federico Rodriguez Hertz, Zhiren Wang

TL;DR

The paper proves that self-conformal measures in the plane exhibit polynomial Fourier decay: there exists $\alpha>0$ and $C>0$ such that $|\mathcal{F}_q(\nu)| \le C|q|^{-\alpha}$ for any self-conformal measure $\nu$ under mild nonlinearity and irreducibility assumptions. The authors develop a novel cocycle-based refinement of Oh–Winter’s Dolgopyat method in the $C^ω$ plane setting, combining a derivative cocycle with an angle cocycle and a disintegration/model framework to achieve a uniform spectral gap for the transfer operator $P_{s,\ell}$. This spectral gap yields an exponentially fast equidistribution for the renormalized cocycle, enabling a renewal theorem with exponential speed and a residue process that together drive the polynomial decay of the Fourier transform. The results extend planar Fourier-decay theory beyond self-similar or strongly separated systems, offering a robust, higher-dimensional Dolgopyat-type toolkit applicable to dynamically defined measures in the plane."

Abstract

Let $Φ$ be a $C^ω(\mathbb{C})$ self-conformal IFS on the plane, satisfying some mild non-linearity and irreducibility conditions. We prove a uniform spectral gap estimate for the transfer operator corresponding to the derivative cocycle and every given self-conformal measure. Building on this result, we establish polynomial Fourier decay for any such measure. Our technique is based on a refinement of a method of Oh-Winter (2017) where we do not require separation from the IFS or the Federer property for the underlying measure.

Spectral gaps and Fourier decay for self-conformal measures in the plane

TL;DR

The paper proves that self-conformal measures in the plane exhibit polynomial Fourier decay: there exists and such that for any self-conformal measure under mild nonlinearity and irreducibility assumptions. The authors develop a novel cocycle-based refinement of Oh–Winter’s Dolgopyat method in the plane setting, combining a derivative cocycle with an angle cocycle and a disintegration/model framework to achieve a uniform spectral gap for the transfer operator . This spectral gap yields an exponentially fast equidistribution for the renormalized cocycle, enabling a renewal theorem with exponential speed and a residue process that together drive the polynomial decay of the Fourier transform. The results extend planar Fourier-decay theory beyond self-similar or strongly separated systems, offering a robust, higher-dimensional Dolgopyat-type toolkit applicable to dynamically defined measures in the plane."

Abstract

Let be a self-conformal IFS on the plane, satisfying some mild non-linearity and irreducibility conditions. We prove a uniform spectral gap estimate for the transfer operator corresponding to the derivative cocycle and every given self-conformal measure. Building on this result, we establish polynomial Fourier decay for any such measure. Our technique is based on a refinement of a method of Oh-Winter (2017) where we do not require separation from the IFS or the Federer property for the underlying measure.
Paper Structure (21 sections, 25 theorems, 243 equations)

This paper contains 21 sections, 25 theorems, 243 equations.

Table of Contents

  1. Introduction
  2. Fourier decay for self-conformal measures on the plane
  3. Spectral gap via a cocycle version of a technique of Oh-Winter, and the $C^2$ case
  4. Spectral gap
  5. Some notations and conventions
  6. Preliminaries
  7. A model (disintegration) construction
  8. Constructing the model
  9. Proof of Part (6) of Theorem \ref{['Theorem disint']}
  10. Statement of spectral gap and reduction to the construction of Dolgopyat operators
  11. Proof of Lemma \ref{['Lemma 5.4 Naud']}
  12. Some preliminary Lemmas
  13. Construction of the Dolgopyat operators
  14. Proof of Part 1 of Lemma \ref{['Lemma 5.4 Naud']}
  15. Proof of Part 2 of Lemma \ref{['Lemma 5.4 Naud']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\Phi$ be a uniformly contracting $C^{\omega} (D)$ IFS, defined on the closure of the unit disc $D=\overline{B_1(0)}$. Assume: Then for every self-conformal measure $\nu$ there exist $\alpha=\alpha(\nu)$ and $C=C(\nu)>0$ satisfying

Theorems & Definitions (44)

  • Theorem 1.1
  • Claim 2.1
  • Claim 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 34 more