Mixing, counting, and equidistribution in Lie groups
type: Item
instance of: scholarly article
title: Mixing, counting, and equidistribution in Lie groups
zbMATH Open document ID: 0798.11025
DOI: 10.1215/S0012-7094-93-07108-6
author: Alex Eskin
author: Curtis T. McMullen
published in: Duke Mathematical Journal
publication date: 1994-11-01T00:00:00Z
review text: The authors discuss results on mixing, counting and equidistribution in general affine symmetric spaces \(V=G/H\), \(G\) a connected semisimple Lie group with finite center, \(H\) a closed subgroup which is the fixed point of some (not necessarily Cartan-) involution. Let \(\Gamma \subset G\) be a lattice (i.e. \(\Gamma\) is discrete, and \(\Gamma \backslash G\) has finite volume), that has dense projection to \(G/G'\) for any positive-dimensional normal noncompact Lie subgroup \(G' \subseteq G\) and such that \(\Gamma \cap H\) is a lattice in \(H\). The following mixing theorem for \(X = \Gamma \backslash G\) is a well known consequence of the Howe-Moore theorem on the decay of matrix coefficients of unitary representations, [see \textit{R. E. Howe} and \textit{C. C. Moore}, J. Funct. Anal. 32, 72-96 (1979; Zbl 0404.22015)]: \(\int \alpha (xg) \beta (x)dx \to (\int_ x \alpha \int_ x \beta)/m(x)\) as \(g\) tends to infinity. The authors prove that the translates \(Yg\) of the \(H\)-orbit \(Y = (\Gamma \cap H) \backslash H\) become equidistributed on \(X = \Gamma \backslash G\). That is \[ {1 \over m (Y)} \int_{Yg} f(h)dh \to {1 \over m(X)} \int_ Xf(x)dx \] as \(g\) leaves compact subsets of \(H \backslash G\). Finally a counting theorem is shown which contains some of the main results of a paper of \textit{W. Duke}, \textit{Z. Rudnick} and \textit{P. Sarnak} [ibid. 71, No. 1, 143-179 (1993); see the review above]. For a sequence of certain well rounded sets \(B_ n\) it is shown that the cardinality of the number of points of \(\Gamma v\) \((v\) denotes the coset \(H\) in \(G/H)\) which lie in \(B_ n\) grows like the volume of \(B_ n\) asymptotically \[ | \Gamma v \cap B_ n | \sim {m \bigl( (\Gamma \cap H) \backslash H \bigr) \over m(\Gamma \backslash G)} m(B_ n). \] The paper starts with interesting examples concerning the hyperbolic plane which essentially already appeared in the thesis of \textit{G. A. Margulis} [Funct. Anal. Appl. 4, 335-336 (1969); translation f…
Mathematics Subject Classification ID: 11G99
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 495441
zbMATH Keywords: mixing
zbMATH Keywords: counting
zbMATH Keywords: equidistribution
zbMATH Keywords: affine symmetric spaces
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: wave front lemma
Wikidata QID: Q101426571
reviewed by: Harald Rindler
MaRDI profile type: Publication
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full work available at URL: s0012-7094-93-07108-6
OpenAlex ID: W2106512332
description: scientific article; zbMATH DE number 495441