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

Abstract

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.

Figures (11)

• Figure 1: An illustration showing the points $u$ and $v$. The dashed arc shows the horoball along which the point $u$ is pulled under the action of $f$.
• Figure 2: Applying Lemma \ref{['Circle']}, considering the dashed horoball.
• Figure 3: The cross ratio visualised.
• Figure 4: An overview of the horoball $H_p$ along with our chosen points. We wish to find a lower bound for $d_\mathbb{H}(z_{T\theta}, v)$.
• Figure 5: Bounding $\vert z - p \vert$ from above using Lemma \ref{['Circle']}.

Theorems & Definitions (16)

• Theorem 2.1
• Theorem 2.2: Global Measure Formula
• Theorem 3.1
• Theorem 3.2
• Lemma 4.1: Circle Lemma
• proof
• Lemma 4.2: Parabolic Centre Lemma
• proof
• Lemma 4.3: Horoball Radius Lemma
• proof