Hölder Dimension with Applications to Hyperbolic Groups

Abstract

We introduce the ‘Hölder dimension’ of a metric space as the infimum of Hausdorff dimensions of Hölder equivalent spaces. This definition is a Hölder variant of the definition of conformal dimension, the study of which serves as inspiration for some of our work.
We present two main results, along with supporting results and examples. Firstly, in Chapter 4, we show that capacity dimension is an upper bound for the Hölder dimension of any compact, doubling metric space with finite capacity dimension. Consequently, for locally self-similar metric spaces, Hölder dimension is equal to topological dimension. We explore some caveats of this result in Chapter 5 where we give examples of Hölder dimension equalling topological dimension but only as a strict infimum, not a minimum, and an example of a space, which is not locally self-similar that has Hölder dimension strictly greater than its topological dimension. Secondly, in Chapter 6, we explain how to reduce understanding the Hölder dimension of boundaries of hyperbolic groups to understanding the Hölder dimension of the boundaries of hyperbolic groups with connected boundary.

Date of Award	29 Sept 2020
Original language	English
Awarding Institution	The University of Bristol
Supervisor	John M Mackay (Supervisor) & Nina C Snaith (Supervisor)