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

TL;DR

This paper analyzes the Assouad spectrum of Kleinian limit sets and Patterson-Sullivan measures, revealing new geometric features and differences from traditional dimensions, enhancing understanding of fractal structures in hyperbolic geometry.

Contribution

It provides a detailed analysis of the Assouad spectrum for Kleinian limit sets and Patterson-Sullivan measures, uncovering novel geometric interactions not seen in classical dimensions.

Findings

Assouad spectrum can exceed Hausdorff and box dimensions.

Interplay between horoballs of different ranks affects the spectrum.

New geometric features are revealed through the spectrum analysis.

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.