We conduct a detailed analysis of the Assouad type dimensions and spectra in the context of limit sets of geometrically finite Kleinian groups and Julia sets of parabolic rational maps. Our analysis includes the Patterson-Sullivan measure in the Kleinian case and the analogous conformal measure in the Julia set case. Our results constitute a new perspective on the Sullivan dictionary between Kleinian groups and rational maps. We show that there exist both strong correspondences and strong differences between the two settings. The differences we observe are particularly interesting since they come from dimension theory, a subject where the correspondence described by the Sullivan dictionary is especially strong.

Parabolic fixed points form a countable dense subset of the limit set of a non-elementary geometrically finite Kleinian group with at least one parabolic element. Given such a group, one may associate a standard set of pairwise disjoint horoballs, each tangent to the boundary at a parabolic fixed point. The diameter of such a horoball can be thought of as the 'inverse cost' of approximating an arbitrary point in the limit set by the associated parabolic point. A result of Stratmann and Velani allows one to count horoballs of a given size and, roughly speaking, for small \(r>0\) there are \(r^{-\delta}\) many horoballs of size approximately \(r\), where \(\delta\) is the Poincaré exponent of the group. We investigate localisations of this result, where we seek to count horoballs of size approximately \(r\) inside a given ball \(B(z,R)\). Roughly speaking, if \(r \lesssim R^2\), then we obtain an analogue of the Stratmann-Velani result (normalised by the Patterson-Sullivan measure of \(B(z,R)\)). However, for larger values of \(r\), the count depends in a subtle way on \(z\).   Our counting results have several applications, especially to the geometry of conformal measures supported on the limit set. For example, we compute or estimate several 'fractal dimensions' of certain \(s\)-conformal measures for \(s>\delta\) and use this to examine continuity properties of \(s\)-conformal measures at \(s=\delta\).

The Assouad dimension of the limit set of a geometrically finite Kleinian group with parabolics may exceed the Hausdorff and box dimensions. The Assouad spectrum is a continuously parametrised family of dimensions which ‘interpolates’ between the box and Assouad dimensions of a fractal set. It is designed to reveal more subtle geometric information than the box and Assouad dimensions considered in isolation. We conduct a detailed analysis of the Assouad spectrum of limit sets of geometrically finite Kleinian groups and the associated Patterson–Sullivan measure. Our analysis reveals several novel features, such as interplay between horoballs of different rank not seen by the box or Assouad dimensions.

The Assouad dimension of the limit set of a geometrically finite Kleinian group with parabolics may exceed the Hausdorff and box dimensions. The Assouad spectrum is a continuously parametrised family of dimensions which 'interpolates' between the box and Assouad dimensions of a fractal set. It is designed to reveal more subtle geometric information than the box and Assouad dimensions considered in isolation. We conduct a detailed analysis of the Assouad spectrum of limit sets of geometrically finite Kleinian groups and the associated Patterson-Sullivan measure. Our analysis reveals several novel features, such as interplay between horoballs of different rank not seen by the box or Assouad dimensions.

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 dimension are distinct. The box and packing dimensions of the Julia set are known to coincide with h and, moreover, h can be characterised by a topological pressure function. This distinctive behaviour of the Assouad dimension invites further analysis of the Assouad type dimensions, including the Assouad and lower spectra. We compute 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.