arXiv subjects

Liam Stuart

Publications and source records attributed to Liam Stuart.

4 recordsLinked to original sources

arXiv2022

The Assouad spectrum of Kleinian limit sets and Patterson-Sullivan measure

The Assouad dimension of the limit set of a geometrically finite Kleinian group with parabolics may exceed the Hausdorff and box dimensions. The Assouad \emph{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.

math.DS↗

arXiv2022

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.

math.DS↗

arXiv2022

Refined horoball counting and conformal measure for Kleinian group actions

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 Poincar\'e 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 $s=\delta$.

math.DS↗

arXiv2020

A new perspective on the Sullivan dictionary via Assouad type dimensions and spectra

The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. An especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. In recent work we established formulae for the Assouad type dimensions and spectra for these fractal sets and certain conformal measures they support. This allows a rather more nuanced comparison of the two families in the context of dimension. In this expository article we discuss how these results provide new entries in the Sullivan dictionary, as well as revealing striking differences between the two settings.

math.DS↗