{"title":"Equidistribution and counting of orbit points for discrete rank one isometry groups of Hadamard spaces","authors":"G. Link","doi":"10.2140/tunis.2020.2.791","DOIUrl":null,"url":null,"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.","PeriodicalId":36030,"journal":{"name":"Tunisian Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2018-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/tunis.2020.2.791","citationCount":"10","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Tunisian Journal of Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/tunis.2020.2.791","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.