Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups
type: Item
instance of: scholarly article
title: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups
zbMATH Open document ID: 0566.58022
DOI: 10.1007/BF02392379
published in: Acta Mathematica
publication date: 1984-01-01T00:00:00Z
review text: Let \(\Gamma\) be a geometrically finite Kleinian group acting in hyperbolic space \({\mathbb{H}}^ 3\) (or more generally \({\mathbb{H}}^ n)\) with limit set \(\Lambda\). This paper continues the study begun earlier [\textit{D. Sullivan}, Publ. Math., Inst. Hautes Etud. Sci. 50, 171-202 (1979; Zbl 0439.30034)] of geometric measures on \(\Lambda\). These are finite measures \(\mu\) which translate by the rule \(\gamma *\mu =| \gamma '|^{\delta}\mu\), \(\gamma\in \Gamma\), where \(\delta =Haus. Dim.(\Lambda)\). (\(\delta\) is also the critical exponent \(\overline{\lim}_{R\to \infty}(1/R)\log n(R)\) of \(\Gamma\), where n(R) is the number of orbit points in a ball of radius R and fixed center.) Such \(\mu\) are constructed following Patterson as a limit of weighted atomic measures on orbits of \(\Gamma\) in \({\mathbb{H}}^ 3\). The author proves that \(\mu\) is uniquely determined by the translation rule and \(\mu (\Lambda)=1.\) Let \(\nu_ c\) denote the Hausdorff (covering) measure of \(\lambda\) with gauge function \(r^{\delta}\). Dually one introduces the packing measure \(\nu_ p(U)=\lim_{\epsilon \to 0} \sup \sum r_ i^{\delta}\), where \(U\subset \Lambda\) is open and the sup is over packings of U by disjoint balls \(B_ i\in U\) of radii \(\leq \epsilon\). Let \(\mu\) (x,r) be the size of a ball of center x, radius r. The inequalities \[ c\leq \mu (x,r)/r^{\delta}\leq_{i.o.}C\quad and\quad c\leq_{i.o.}\mu (x,r)/r^{\delta}\leq C \] (where i.o. means for a sequence \(r_ i\to 0)\) imply respectively that \(\mu =\nu_ p\) and \(\mu =\nu_ c.\) Suitable estimates show that if \(\Gamma\) has no cusps then \(\mu =\nu_ p=\nu_ c\). If \(\Gamma\) has cusps the situation is more complicated: one may have \(\mu =\nu_ c\neq \nu_ p\) (rank 1 cusps, \(1<\delta <2)\) or \(\mu =\nu_ p\neq \nu_ c\) (rank 1 cusps, \(\delta <1)\). The only unresolved case is when \(1<\delta <2\) and \(\Gamma\) has cusps of ranks 1 and 2. Then \(0=\nu_ c\neq \nu_ p=\infty\) and \(\mu\) is unidentified. …
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 54F50
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 11F06
zbMATH DE Number: 3903608
zbMATH Keywords: entropy
zbMATH Keywords: geodesic flow
zbMATH Keywords: geometrically finite Kleinian group
zbMATH Keywords: limit set
Wikidata QID: Q105982858
MaRDI profile type: Publication
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author: Dennis Sullivan
description: scientific article; zbMATH DE number 3903608