Harmonic measures for compact negatively curved manifolds
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.
scientific article; zbMATH DE number 1143397
scientific article