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Fourier dimension of Mandelbrot Cascades on planar curves

Donggeun Ryou, Ville Suomala

Abstract

We consider multifractal Mandelbrot cascades supported on planar $C^2$ curves with nonvanishing curvature and show that their Fourier dimension is as large as possible, i.e., equal to the infimum of the lower pointwise dimension of the measure.

Fourier dimension of Mandelbrot Cascades on planar curves

Abstract

We consider multifractal Mandelbrot cascades supported on planar curves with nonvanishing curvature and show that their Fourier dimension is as large as possible, i.e., equal to the infimum of the lower pointwise dimension of the measure.

Paper Structure

This paper contains 9 sections, 18 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation for cascade measures
  4. Auxiliary results for the Mandelbrot cascades
  5. A concentration inequality
  6. The lower bound of the Fourier dimension
  7. The upper bound of the Fourier dimension
  8. Decay of the spherical average
  9. Appendix

Key Result

Theorem 1.1

$\dim_F\mu=\alpha_{\min}$ almost surely on non-extinction.

Theorems & Definitions (36)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Lemma 2.5
  • proof
  • ...and 26 more