Entropy and escape of mass for Hilbert modular spaces
type: Item
instance of: scholarly article
zbMATH Open document ID: 1276.37004
publication date: 2012-08-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1101.4658
full work available at URL: jlt22029.htm
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 28D20
Mathematics Subject Classification ID: 22D40
zbMATH DE Number: 6072203
zbMATH Keywords: escape of mass
zbMATH Keywords: entropy
zbMATH Keywords: diagonal actions
zbMATH Keywords: \({\mathbb Q}\)-rank one
author: S. Kadyrov
MaRDI profile type: Publication
arXiv ID: 1101.4658
arXiv classification: math.DS
title: Entropy and escape of mass for Hilbert modular spaces
published in: Journal of Lie Theory
review text: In dynamical study of diagonal flows, one considers a sequence of probability measures invariant under a diagonal element of a linear group acting on a homogeneous space. It is known that the limit measure of the space could be any value in the unit interval \([0, 1]\). In this paper under review, the author studies the relation between metric entropy and escape of mass for Hilbert modular spaces with the action of a diagonal element.NEWLINENEWLINETo state the result, let \(F\) be an algebraic number field and let \({\mathcal O}\) be its ring of integers. Let \(S^{\infty}= \{\sigma_1,\ldots, \sigma_{r+s}\}\) be its Archimedean places where \(\{\sigma_1,\ldots, \sigma_r\}\) are the real places and the rest are complex ones. Define \(G := \prod_{n=1}^r \mathrm{SL}_2\left({\mathbb R}\right) \times \prod_{m=1}^{s}\mathrm{SL}_2 \left({\mathbb C}\right)\) and \(\Gamma := \mathrm{SL}_2\left({\mathcal O}\right).\) Then, \(\Gamma\) is naturally embedded into \(G\) via \(\Delta(\gamma) = \left(\sigma_1(\gamma), \ldots, \sigma_{r+s}(\gamma)\right)\) for \(\gamma\in \Gamma.\) Set \(X := \Gamma \backslash G\). Let \(a\) be any fixed diagonal element of \(G\) given by \(a = \mathrm{diag}(e^{i\theta_1}e^{a_1/2}, e^{-i\theta_1}e^{-a_1/2})\times \cdots\times \mathrm{diag}(e^{i\theta_{r+s}}e^{a_{r+s}/2}, e^{-i\theta_{r+s}}e^{-a_{r+s}/2})\) with \(a_j\in {\mathbb R}\), \(\theta_j\in [0, 2 \pi]\) and \(\theta_1, \ldots, \theta_r = 0.\) Define the action \(T\) on \(X\) by \(T(x) = x\cdot a.\)NEWLINENEWLINEFor a positive constant \(M\), define \(X_{< M} = \{x\in X: \mathrm{ht}(x) < M\}\) where \(\mathrm{ht}(\cdot)\) is called the height funtion on \(X\) defined in the paper. Let \(h_{\max}(T)\) denote the maximal metric entropy of \(T\). Then, one of the main results can be stated as follows. For \(M > \max\{e^{3 h_{\max}(T)}, 100\}\) be given. Then there exists a continuous decreasing function \(\phi : {\mathbb R}^{+}\to {\mathbb R}\) with \(\lim_{M\to\infty} \phi(M) = 0\) such that \(\…
reviewed by: Liang-Chung Hsia
description: scientific article; zbMATH DE number 6072203