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.
Categories: math.DS math.CA