Regularity of Kleinian limit sets and Patterson-Sullivan measures
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- by Jonathan M. Fraser PDF
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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.References
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Additional Information
- Jonathan M. Fraser
- Affiliation: School of Mathematics and Statistics, The University of St Andrews, St Andrews, KY16 9SS, Scotland
- MR Author ID: 946983
- Email: jmf32@st-andrews.ac.uk
- 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).
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 4977-5009
- MSC (2010): Primary 30F40, 28A80; Secondary 37F30, 37C45
- DOI: https://doi.org/10.1090/tran/7830
- MathSciNet review: 4009399