Julia sets of hyperbolic rational maps have positive Fourier dimension

TL;DR

This paper proves that Julia sets of hyperbolic rational maps have positive Fourier dimension, and shows that many measures supported on these sets exhibit polynomial Fourier decay, advancing understanding of their harmonic analysis properties.

Contribution

It establishes positive Fourier dimension for Julia sets of hyperbolic rational maps and demonstrates polynomial Fourier decay for a broad class of measures supported on these sets.

Findings

Julia sets have positive Fourier dimension

Measures on Julia sets exhibit polynomial Fourier decay

Results apply to measures of maximal entropy and conformal measures

Abstract

Let $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ be a hyperbolic rational map of degree $d \geq 2$, and let $J \subset \mathbb{C}$ be its Julia set. We prove that $J$ always has positive Fourier dimension. The case where $J$ is included in a circle follows from a recent work of Sahlsten and Stevens, see arXiv:2009.01703. In the case where $J$ is not included in a circle, we prove that a large family of probability measures supported on $J$ exhibit polynomial Fourier decay: our result applies in particular to the measure of maximal entropy and to the conformal measure.