Limit sets, Julia sets and Sullivan’s dictionary : a dimension theoretic analysis

Abstract

This thesis includes work from four papers that were written during the author’s time as a PhD student with Jonathan Fraser, namely [40, 41, 42, 43]. Chapter 1 introduces the two main settings that will be studied throughout this thesis along with several tools that will be used. This will include various notions of dimensions of sets and measures, the setting of hyperbolic geometry and limit sets, and the setting of rational maps and Julia sets. Chapter 2 will state and prove results in the hyperbolic geometry setting, where we calculate the Assouad and lower spectra for limit sets of geometrically finite Kleinian groups along with their associated Patterson-Sullivan measure. The broad approach takes some ideas from [35] where the Assouad and lower dimensions were calculated, but many of the ideas require adjustment or replacement due to the Assouad and lower spectra requiring finer control. An important tool made use of is the notion of a ‘global measure formula’ in order to obtain estimates on efficient covers. Chapter 3 involves adapting this approach to calculate the Assouad type dimensions of Julia sets of parabolic rational maps and their associated h-conformal measures, where h denotes the Hausdorff dimension of the Julia set. Chapter 4 is then dedicated to a discussion of the results in Chapters 2 and 3 in the context of Sullivan’s dictionary, a framework which draws many connections between the settings of hyperbolic geometry and rational maps. We draw several interesting parallels between the two settings, along with some notable differences, using our results on the Assouad type dimensions which are not witnessed by other notions of dimension. In Chapter 5, we obtain results for counting horoballs of certain sizes, and then discuss some applications of these results to Diophantine approximation and the calculation of dimensions of conformal measures. We finish in Chapter 6 with discussion about further questions which stem from our research.