Constructing Restricted Patterson Measures for Geometrically Infinite Kleinian Groups

作     者：Kurt FALK Bernd O. STRATMANN 

作者机构：Mathematical Institute University of Bern Mathematical Institute University of St Andrews North Haugh St Andrews KY16 9SS Scotland UK 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2006年第22卷第2期

页      面：431-446页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Research supported by the Schweizer Nationalfonds No. 20-61379.00 European Project TMR "Geometric Analysis" ACR-OFES No. UE 00.0349 

主　　题：Kleinian group Patterson measure Hausdorff dimension 

摘      要：In this paper, we study exhaustions, referred to as p-restrictions, of arbitrary nonelementary Kleinian groups with at most finitely many bounded parabolic elements. Special emphasis is put on the geometrically infinite case, where we obtain that the limit set of each of these Kleinian groups contains an infinite family of closed subsets, referred to as p-restricted limit sets, such that there is a Poincaré series and hence an exponent of convergence δp, canonically associated with every element in this family. Generalizing concepts which are well known in the geometrically finite case, we then introduce the notion of p-restricted Patterson measure, and show that these measures are non-atomic, δp-harmonic, δp-subconformal on special sets and δp-conformal on very special sets. Furthermore, we obtain the results that each p-restriction of our Kleinian group is of δp-divergence type and that the Hausdorff dimension of the p-restricted limit set is equal to δp.