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Fourier transforms and iterated function systems

Tuomas Sahlsten

TL;DR

This survey investigates how the Fourier transforms of stationary measures for iterated function systems decay, linking pseudo-randomness from arithmetic, algebraic, and geometric structure to the behavior of $\widehat{\mu}(\xi)$. It canvasses techniques from thermodynamic formalism, additive combinatorics, renewal theory, and random walks on groups, spanning self-similar, self-conformal, and higher-dimensional IFSs, and highlights obstructions such as Pisot and Salem numbers that govern decay. Key results include dichotomies and polylogarithmic or polynomial decay in equicontractive Bernoulli convolutions, polynomial decay in non-linear self-conformal cases (under UNI and related conditions), and subpolynomial to polynomial decay in higher dimensions under rotation and affine-group dynamics, with Fractal Uncertainty Principles providing a bridge to quantum-chaos questions. The work also outlines major open problems, including sharp decay rates under overlaps, full nonlinear high-dimensional theories, and potential finite-field analogues, underscoring the deep connections between fractal geometry, harmonic analysis, and dynamical systems.

Abstract

We discuss the problem of bounding the Fourier transforms of stationary measures of iterated function systems (IFSs) and how the pseudo-randomness of the IFS either due to arithmetic, algebraic or geometric reasons is reflected in the behaviour of the Fourier transform. We outline various methods that have been built to estimate the Fourier transform of stationary measures arising e.g. from thermodynamical formalism, additive combinatorics, random walks on groups and hyperbolic dynamics. Open problems, prospects and recent links to quantum chaos are also highlighted.

Fourier transforms and iterated function systems

TL;DR

This survey investigates how the Fourier transforms of stationary measures for iterated function systems decay, linking pseudo-randomness from arithmetic, algebraic, and geometric structure to the behavior of . It canvasses techniques from thermodynamic formalism, additive combinatorics, renewal theory, and random walks on groups, spanning self-similar, self-conformal, and higher-dimensional IFSs, and highlights obstructions such as Pisot and Salem numbers that govern decay. Key results include dichotomies and polylogarithmic or polynomial decay in equicontractive Bernoulli convolutions, polynomial decay in non-linear self-conformal cases (under UNI and related conditions), and subpolynomial to polynomial decay in higher dimensions under rotation and affine-group dynamics, with Fractal Uncertainty Principles providing a bridge to quantum-chaos questions. The work also outlines major open problems, including sharp decay rates under overlaps, full nonlinear high-dimensional theories, and potential finite-field analogues, underscoring the deep connections between fractal geometry, harmonic analysis, and dynamical systems.

Abstract

We discuss the problem of bounding the Fourier transforms of stationary measures of iterated function systems (IFSs) and how the pseudo-randomness of the IFS either due to arithmetic, algebraic or geometric reasons is reflected in the behaviour of the Fourier transform. We outline various methods that have been built to estimate the Fourier transform of stationary measures arising e.g. from thermodynamical formalism, additive combinatorics, random walks on groups and hyperbolic dynamics. Open problems, prospects and recent links to quantum chaos are also highlighted.
Paper Structure (14 sections, 36 theorems, 105 equations, 1 figure)

This paper contains 14 sections, 36 theorems, 105 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Self-similar iterated function systems in $\mathbb{R}$
  3. Equicontractive systems and Bernoulli convolutions
  4. General self-similar systems
  5. Non-linear self-conformal iterated function systems in $\mathbb{R}$
  6. Fourier decay via nonlinearity of the Gauss map
  7. Additive combinatoric approach and Fuchsian groups
  8. Renewal theoretic approach and random walks on groups
  9. General non-linear iterated function systems
  10. Higher dimensions
  11. Self-similar measures and random walks on rotations
  12. Self-affine measures and random walks on affine groups
  13. Non-linear $C^{1+\alpha}$ systems in $\mathbb{R}^d$ and Fractal Uncertainty Principles
  14. Prospects

Key Result

Theorem 2.1

$\widehat{\mu}_\lambda(\xi) \to 0$ as $|\xi| \to \infty$ if and only if $\lambda^{-1}$ is not a Pisot number or $\lambda = 1/2$.

Figures (1)

  • Figure 1: Left: Graph of $|\widehat{\mu}(\xi)|$ for a self-similar measure (Bernoulli convolution) associated to the IFS $\{x \mapsto \lambda x - 1,x \mapsto \lambda x + 1\},$ where $\lambda = (\frac{\sqrt{5}+1}{2})^{-2}$. Here $\lambda^{-2}$ is a Pisot number, so there is no Fourier decay for $\mu$, see Theorem \ref{['thm:erdossalem']} below. Right: Graph of $|\widehat{\mu}(\xi)|$ for a self-conformal measure associated to the non-linear IFS $\{x \mapsto \tfrac{x}{3} + \tfrac{x^{2}}{6},x \mapsto \tfrac{x}{3} + \tfrac{2}{3}\}.$ The non-linearity of the IFS guarantees that there is polynomial Fourier decay for $\mu$, see Theorem \ref{['thm:StevensSahlsten']} below. The orange dotted line in both pictures is the polynomial function $|\xi|^{-\alpha/2}$ for reference, where $\alpha = 0.6 \approx \dim_\mathrm{H} \mu$, the Hausdorff dimension of $\mu$Falconer, in both cases. This suggests the second measure is potentially close to being a Salem measure: $|\widehat{\mu}(\xi)| \lesssim |\xi|^{-\frac{\dim_\mathrm{H} \mu}{2}}$ as $|\xi| \to \infty$. Pictures by C. Wormell W1W1aW2.

Theorems & Definitions (43)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.9
  • Theorem 2.10
  • Theorem 2.11
  • Theorem 2.12
  • proof : Sketch of the proof of Theorem \ref{['thm:lisahlsten']}
  • ...and 33 more