Extreme instability of the horocycle flow
type: Item
instance of: scholarly article
title: Extreme instability of the horocycle flow
zbMATH Open document ID: 0773.58019
DOI: 10.1007/BF03041065
author: Serge Troubetzkoy
published in: Journal d'Analyse Mathématique
publication date: 1993-06-29T00:00:00Z
review text: Let \(S\) be a compact oriented surface of genus \(\geq 2\) and let \(g\) be a \(C^ 3\)-Riemannian metric on \(S\) of negative curvature. A uniformized version of a result of Feldman and Ornstein for the horocycle flow \(h_ t\) on the unit tangent bundle \(T^ 1S\) of \(S\) with its Margulis- parametrization is presented: Namely there is a number \(\beta>0\) only depending on \(g\) such that for the horocycle flow \(\widehat h_ t\) induced by any metric \(\widehat g\) on \(S\) which is sufficiently close to \(g\) in the \(C^ 2\)-topology and for any two points \(x,y\in T^ 1S\) with \(\text{dist}(x,y)<\beta\) and which are not contained in a single \(\widehat h_ t\)-orbit it holds that \[ \lim_{S\to\infty}{1\over S}\int^ S_ 0\text{dist}(\widehat h_ tx,h_ ty)dt\geq\beta. \] This result is used to show that in contrast to the structural stability of the geodesic flow of \(g\) the horocycle flow \(h_ t\) is instable under small perturbations of \(g\). For \(\widehat g\) close to \(g\) a distance \(\overline d(h_ t,\widehat h_ t)\) between the corresponding horocycle flows \(h_ t\), \(\widehat h_ t\) on \(T^ 1S\) can be defined via stationary coupling which measures the difference in long time behaviour of typical orbits of the flows \(h_ t\), \(\widehat h_ t\) through nearby points. It is shown that \(g(h_ t,\widehat h_ t)\geq A>0\) for a universal constant \(A\) whenever \(\widehat h_ t\) is the horocycle flow of a normalized metric \(\widehat g\) on \(S\) which is sufficiently close to \(g\).
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
zbMATH DE Number: 221304
zbMATH Keywords: horocycle flows
zbMATH Keywords: structural instability
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03041065
OpenAlex ID: W159783938
description: scientific article; zbMATH DE number 221304