Equidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaces

IF 0.8 Q2 MATHEMATICS
G. Link
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引用次数: 10

Abstract

Let $X$ be a proper, geodesically complete Hadamard space, and $\ \Gamma<\mbox{Is}(X)$ a discrete subgroup of isometries of $X$ with the fixed point of a rank one isometry of $X$ in its infinite limit set. In this paper we prove that if $\Gamma$ has non-arithmetic length spectrum, then the Ricks' Bowen-Margulis measure -- which generalizes the well-known Bowen-Margulis measure in the CAT$(-1)$ setting -- is mixing. If in addition the Ricks' Bowen-Margulis measure is finite, then we also have equidistribution of $\Gamma$-orbit points in $X$, which in particular yields an asymptotic estimate for the orbit counting function of $\Gamma$. This generalizes well-known facts for non-elementary discrete isometry groups of Hadamard manifolds with pinched negative curvature and proper CAT$(-1)$-spaces.
Hadamard空间离散秩一等距群轨道点的均分与计数
设$X$是一个固有的、测地完备的Hadamard空间,$\ \Gamma<\mbox{Is}(X)$是$X$等距的离散子群,其无限极限集中$X$的一个秩一等距的不动点。在本文中,我们证明了如果$\Gamma$具有非算术长度谱,那么Ricks' Bowen-Margulis测度——它推广了CAT$(-1)$设置中的著名的Bowen-Margulis测度——是混合的。另外,如果Ricks' bowwen - margulis测度是有限的,那么我们也有$\Gamma$-轨道点在$X$上的均匀分布,这特别地产生了$\Gamma$的轨道计数函数的渐近估计。这推广了具有缩紧负曲率和固有CAT$(-1)$-空间的非初等离散Hadamard流形等距群的已知事实。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Tunisian Journal of Mathematics
Tunisian Journal of Mathematics Mathematics-Mathematics (all)
CiteScore
1.70
自引率
0.00%
发文量
12
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