Regularity of Kleinian limit sets and Patterson-Sullivan measures
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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.
Fraser , J M 2019 , ' Regularity of Kleinian limit sets and Patterson-Sullivan measures ' , Transactions of the American Mathematical Society , vol. 372 , no. 7 , pp. 4977-5009 . https://doi.org/10.1090/tran/7830
Transactions of the American Mathematical Society
Copyright © 2019, American Mathematical Society. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1090/tran/7830
DescriptionFunding: Leverhulme Trust Research Fellowship (RF-2016-500).
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