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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Regularity of Kleinian limit sets and Patterson-Sullivan measures


Author: Jonathan M. Fraser
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 30F40, 28A80; Secondary 37F30, 37C45
DOI: https://doi.org/10.1090/tran/7830
Published electronically: June 21, 2019
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Abstract: We consider several (related) notions of geometric regularity in the context of limit sets of geometrically finite Kleinian groups and associated Patterson-Sullivan measures. We begin by computing the upper and lower regularity dimensions of the Patterson-Sullivan measure, which involves controlling the relative measure of concentric balls. We then compute the Assouad and lower dimensions of the limit set, which involves controlling local doubling properties. Unlike the Hausdorff, packing, and box-counting dimensions, we show that the Assouad and lower dimensions are not necessarily given by the Poincaré exponent.


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Jonathan M. Fraser
Affiliation: School of Mathematics and Statistics, The University of St Andrews, St Andrews, KY16 9SS, Scotland
Email: jmf32@st-andrews.ac.uk

DOI: https://doi.org/10.1090/tran/7830
Keywords: Kleinian group, Patterson-Sullivan measure, Assouad dimension, regularity dimension
Received by editor(s): May 10, 2018
Received by editor(s) in revised form: February 15, 2019
Published electronically: June 21, 2019
Additional Notes: The author was supported by a Leverhulme Trust Research Fellowship (RF-2016-500).
Article copyright: © Copyright 2019 American Mathematical Society