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

TL;DR

This paper explores the Sullivan dictionary's correspondence between Kleinian groups and rational maps, focusing on Assouad dimensions and spectra to reveal nuanced differences and new insights into their fractal limit and Julia sets.

Contribution

It introduces formulae for Assouad type dimensions and spectra for limit and Julia sets, providing a new perspective and detailed comparison within the Sullivan dictionary framework.

Findings

Established formulae for Assouad dimensions and spectra

Revealed striking differences between limit sets and Julia sets

Provided new entries in the Sullivan dictionary

Abstract

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.