Harmonic measures for compact negatively curved manifolds
type: Item
instance of: scholarly article
title: Harmonic measures for compact negatively curved manifolds
zbMATH Open document ID: 0899.58031
DOI: 10.1007/BF02392709
author: Ursula Hamenstädt
published in: Acta Mathematica
publication date: 1998-11-15T00:00:00Z
review text: Let \(M\) be an \(n\)-dimensional compact Riemannian manifold of negative sectional curvature. The geodesic flow \(\Phi^t\) is a smooth dynamical system on the unit tangent bundle \(T^1M\) of \(M\), generated by the geodesic spray \(X\). It is known that \(T^1M\) admits four natural foliations \(W^{su}\), \(W^u\), \(W^s\), \(W^{ss}\) which are invariant under the geodesic flow. The leaf \(W^{ss}(v)\) containing \(v\in T^1M\) of the strong stable foliation \(W^{ss}\) consists of all points \(w\in T^1M\) with the property that the distance between \(\Phi^tw\) and \(\Phi^tv\) converges to zero as \(t\to\infty\) (where we may use the distance on \(T^1M\) induced by the Sasaki metric). The leaf \(W^s(v)\) through \(v\) of the stable foliation \(W^s\) is \(W^s(v)= \bigcup_{t\in R}\Phi^tW^{ss}(v)\), and the strong unstable foliation \(W^{su}\) (or the unstable foliation \(W^u\)) is the image of \(W^{ss}\) (or \(W^s\)) under the flip map \(F: w\to-w\). The leaf \(W^i(v)\) of \(W^i\) \((i= ss,su,s,u)\) is a smoothly immersed submanifold of \(T^1M\) depending continuously on \(v\) in the \(C^\infty\)-topology. Moreover, the tangent bundle \(TW^i\) of \(W^i\) is a Hölder-continuous subbundle of \(TT^1M\). Definition. A different operator subordinate to \(W^s\) is a differential operator \(L\) on \(T^1M\) with continuous coefficients and such that for every smooth function \(\alpha\) on \(T^1M\) the value of \(L\alpha\) at \(v\in T^1M\) only depends on the restriction of \(\alpha\) to \(W^s(v)\). The purpose of this paper is to investigate ergodic and analytic properties of second-order differential operators \(L\) on \(T^1M\) with Hölder-continuous coefficients and without zero-order terms which are subordinate to the stable foliation \(W^s\). The main results are given by Theorem A and Theorem B announced in introduction.
Mathematics Subject Classification ID: 37A99
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 37C85
zbMATH DE Number: 1143397
zbMATH Keywords: compact negatively curved manifolds
zbMATH Keywords: harmonic measure
zbMATH Keywords: geodesic flow
zbMATH Keywords: stable foliation
reviewed by: Neculai Papaghiuc
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1143397