Julia Sets of Hyperbolic Rational Maps have Positive Fourier Dimension

Abstract

Let \(f:\widehat{{\mathbb {C}}}\rightarrow \widehat{{\mathbb {C}}}\) 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 (Fourier transform and expanding maps on Cantor sets preprint. arXiv:2009.01703, 2020). 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.