Skinning measures in negative curvature and equidistribution of equidistant submanifolds
Let \(M\) denote a complete Riemannian manifold with sectional curvature \(K \leq -1\), and let \(\pi : \tilde{M} \rightarrow M\) denote the universal Riemannian cover. Let \(\Gamma\) be a non-elementary discrete group of isometries of \(\tilde{M}\). Fixing a base point \(x_{0} \in \tilde{M}\) one defines the critical exponent \(\delta_{\Gamma} = \lim_{n \rightarrow \infty} 1/n \log |\{\gamma \in \Gamma : d(x_{0}, \gamma(x_{0})) \leq n \}|\). The constant \(\delta_{\Gamma}\) is independent of the base point \(x_{0}\), and in this article \(\delta_{\Gamma}\) is assumed to be finite. The Bowen-Margulis measure \(m_{BM}\) on \(T^{1}M\) is also assumed to be finite, and in this case it may be described, when normalized, as the unique probability measure of maximal entropy for the geodesic flow on \(T^{1}M\).NEWLINENEWLINEFor a proper closed convex subset \(C\) of \(\tilde{M}\) let \(\partial^{1}_{+}C\) denote the set of outward pointing unit vectors with base points in \(C\). For \(t > 0\) let \(\Sigma_{t} = g^{t}(\partial^{1}_{+}C)\), where \(g^{t}\) is the geodesic flow on \(T^{1}\tilde{M}\). One may define a skinning measure \(\tilde{\sigma}\) on \(\partial^{1}_{+}C\) and \(\Sigma_{t}, t > 0\), with the following properties: NEWLINENEWLINE1) If \(\gamma \in \Gamma\), then \(\gamma_{*} \tilde{\sigma}_{C} = \tilde{\sigma}_{\gamma(C)}\). NEWLINENEWLINE2) If \(s \geq 0\), then \((g^{s})_{*}(\tilde{\sigma}_{C}) = e^{- s \delta_{\Gamma}} \tilde{\sigma}_{C_{s}}\), where \(C_{s}\) is the set of base points in \(\tilde{M}\) of \(\Sigma_{s}\). NEWLINENEWLINE3) The support of \(\tilde{\sigma}_{C} = \{w \in \partial^{1}_{+}C : \gamma_{w}(\infty) \in \Lambda \Gamma \}\), where \(\gamma_{w}(\infty)\) is the point in \(\partial_{\infty} \tilde{M}\) determined by the geodesic \(\gamma_{w}\) and \(\Lambda \Gamma\) denotes the limit set of \(\Gamma\) in \(\partial_{\infty} \tilde{M}\). NEWLINENEWLINEThe equivariance property 1) allows one to define skinning measures on the projections to \(T^{1}M\) of \(\partial^{1}_{+}C\) and \(\Sigma_{t}\).NEWLINENEWLINEA simplified version of the main result states that if \(m_{BM}\) is finite and mixing, then the normalized skinning measures on \(\Sigma_{t}\) equidistribute as \(t \rightarrow \infty\) to the normalized Bowen-Margulis measure on \(T^{1}M\). Moreover, given estimates on the rate of mixing of the geodesic flow the authors obtain estimates on the rate of this equidistribution.
scientific article; zbMATH DE number 6359413
scientific article