Assouad type dimensions of parabolic Julia sets

We prove that the Assouad dimension of a parabolic Julia set is $\max\{1,h\}$ where $h$ is the Hausdorff dimension of the Julia set. Since $h$ may be strictly less than 1, this provides examples where the Assouad and Hausdorff dimensions are distinct. The box and packing dimensions of the Julia set are also known to coincide with $h$ and, moreover, $h$ can be characterised by a topological pressure function. The distinctive behaviour of the Assouad dimension invites further analysis of the `Assouad type dimensions', including the lower dimension and the Assouad and lower spectra. We derive formulae for all of the Assouad type dimensions for parabolic Julia sets and the associated $h$-conformal measure. Further, we show that if a Julia set has a Cremer point, then the Assouad dimension is 2.