Regularity of Kleinian limit sets and Patterson-Sullivan measures

17 Oct 2019  ·  Fraser Jonathan M. ·

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 Poincar\'e exponent. read more

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Dynamical Systems Classical Analysis and ODEs Group Theory Metric Geometry