Regularity of Kleinian limit sets and Patterson-Sullivan measures

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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.
Original languageEnglish
Pages (from-to)4977-5009
Number of pages33
JournalTransactions of the American Mathematical Society
Volume372
Issue number7
Early online date21 Jun 2019
DOIs
Publication statusPublished - 1 Oct 2019

Keywords

  • Kleinian group
  • Patterson-Sullivan measure
  • Assouad dimension
  • Regularity dimension

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