Julia Sets of Hyperbolic Rational Maps have Positive Fourier Dimension

Let $$f:\widehat{{\mathbb {C}}}\rightarrow \widehat{{\mathbb {C}}}$$ be a hyperbolic rational map of degree $$d \ge 2$$ , and let $$J \subset {\mathbb {C}}$$ its Julia set. We prove that J always has positive Fourier dimension. The case where is included in circle follows from recent work Sahlsten Stevens (Fourier transform expanding maps on Cantor sets preprint. arXiv:2009.01703 2020). In the not circle, we large family probability measures supported exhibit polynomial decay: our result applies particular to measure maximal entropy conformal measure.