Limit sets, Julia sets and Sullivan’s dictionary : a dimension theoretic analysis
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13/06/2023Author
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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.
Type
Thesis, PhD Doctor of Philosophy
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