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Abstract
Parabolic fixed points form a countable dense subset of the limit set of a nonelementary geometrically finite Kleinian group with at least one parabolic element. Given such a group, one may associate a standard set of pairwise disjoint horoballs, each tangent to the boundary at a parabolic fixed point. The diameter of such a horoball can be thought of as the ‘inverse cost’ of approximating an arbitrary point in the limit set by the associated parabolic point. A result of Stratmann and Velani allows one to count horoballs of a given size and, roughly speaking, for small r >0 there are r^{−δ} many horoballs of size approximately r, where δ is the Poincaré exponent of the group. We investigate localisations of this result, where we seek to count horoballs of size approximately r inside a given ball B(z, R). Roughly speaking, if r ≲ R^{2}, then we obtain an analogue of the Stratmann–Velani result (normalised by the Patterson–Sullivan measure of B(z, R)). However, for larger values of r, the count depends in a subtle way on z. Our counting results have several applications, especially to the geometry of conformal measures supported on the limit set. For example, we compute or estimate several ‘fractal dimensions’ of certain sconformal measures for s > δ and use this to examine continuity properties of sconformal measures at s = δ.
Original language  English 

Pages (fromto)  325344 
Number of pages  20 
Journal  Annales Academiae Scientiarum FennicaeMathematica 
Volume  48 
Issue number  1 
DOIs  
Publication status  Published  8 May 2023 
Keywords
 Kleinian group
 Parabolic fixed point
 PattersonSullivan measure
 Conformal measure
 Horoballs
 Global measure formula
 Assouad spectrum
 Box dimension
 Diophantine approximation
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Dive into the research topics of 'Refined horoball counting and conformal measure for Kleinian group actions'. Together they form a unique fingerprint.Projects
 3 Finished

Sabbatical Research Grant 2021: Sabbatical Research Grant
Fraser, J. (PI)
1/08/21 → 31/07/22
Project: Standard

New perspectives in the dimension: New perspectives in the dimension theory of fractals
Fraser, J. (PI)
1/09/19 → 31/01/23
Project: Standard

Fourier analytic techniques: Fourier analytic techniques in geometry and analysis
Fraser, J. (PI) & Falconer, K. J. (CoI)
1/02/18 → 11/06/21
Project: Standard