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

Jonathan M. Fraser, Liam Stuart

Published 2020-07-30Version 1

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.