Julia sets of hyperbolic rational maps have positive Fourier dimension

Abstract

Let $f:\hat{\mathbb{C}}\to \hat{\mathbb{C}}$ be a hyperbolic rational map of degree $d\ge 2$, 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.

arXiv comment35 pages. Modified the argument for the upper regularity of equilibrium states in the conformal setting

References in corpus (4)

• Fourier dimension and spectral gaps for hyperbolic surfaces
• Fourier transform of self-affine measures
• Kleinian Schottky groups, Patterson-Sullivan measures and Fourier decay
• Prime number theorems and holonomies for hyperbolic rational maps