Weak singularity spectra of the Patterson measure for geometrically finite Kleinian groups with parabolic elements
type: Item
instance of: scholarly article
title: Weak singularity spectra of the Patterson measure for geometrically finite Kleinian groups with parabolic elements
zbMATH Open document ID: 0961.30033
author: Bernd O. Stratmann
published in: Michigan Mathematical Journal
publication date: 2001-01-01T00:00:00Z
review text: Let \(G\) be a geometrically finite Kleinian group with parabolic elements and \(L(G)\) its limit set. Further let denote \(B(\xi,r)\) the Euclidean ball of radius \(r\) centered at \(\xi\) and \(\mu\) the Patterson measure supported on the limit set \(L(G)\). Let us put for \(\vartheta>0\) \[ I^\vartheta (\mu):=\left \{\xi \in L(G); \liminf_{r\to 0}{\log\mu \bigl(B(\xi,r) \bigr)\over\log r}\leq \vartheta \right\}. \tag{1} \] By turning the inequality in (1), the set \(I_\vartheta (\mu)\) is obtained and changing \(\underline {\text{lim sup}}\) instead \(\underline {\text{lim inf}}\) we obtain further two sets \(S^\vartheta (\mu)\) and \(S_{\vartheta} (\mu)\). In the paper the Hausdorff dimension of these four sets is determined.
reviewed by: A. Klíč
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 37A15
Mathematics Subject Classification ID: 37E30
Mathematics Subject Classification ID: 37E35
zbMATH DE Number: 1548326
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: exponent of convergence
zbMATH Keywords: Jarník limit sets
zbMATH Keywords: well-approximable
zbMATH Keywords: irrationals
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1030132480
OpenAlex ID: W2061692375
DOI: 10.1307/MMJ/1030132480
description: scientific article; zbMATH DE number 1548326