Skinning measures in negative curvature and equidistribution of equidistant submanifolds
type: Item
instance of: scholarly article
title: Skinning measures in negative curvature and equidistribution of equidistant submanifolds
zbMATH Open document ID: 1321.37029
author: Jouni Parkkonen
author: Frédéric Paulin
published in: Ergodic Theory and Dynamical Systems
publication date: 2014-10-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.6398
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 37A25
zbMATH DE Number: 6359413
zbMATH Keywords: negative sectional curvature
zbMATH Keywords: Bowen-Margulis measure
zbMATH Keywords: skinning measure
zbMATH Keywords: critical exponent
zbMATH Keywords: mixing of the geodesic flow
zbMATH Keywords: equidistribution
MaRDI profile type: Publication
OpenAlex ID: W4302406403
Wikidata QID: Q110021986
arXiv ID: 1202.6398
cites work: Topologie du feuilletage fortement stable. (Topology of the strong stable foliation)
cites work: Poincaré series of finite geometric groups
cites work: Remarques sur le spectre des longueurs d'une surface et comptages
cites work: A characterization of Newtonian functions with zero boundary values
cites work: Démonstration de la conjecture \(\tau\). (Proof of the \(\tau\)-conjecture.)
cites work: On the almost sure spiraling of geodesics in negatively curved manifolds
cites work: Counting orbit points in coverings of negatively curved manifolds and Hausdorff dimension of cusp excursions
cites work: Periodic Orbits for Hyperbolic Flows
cites work: On the rigidity of discrete isometry groups of negatively curved spaces
cites work: Geometrical finiteness with variable negative curvature
cites work: A new description of the Bowen—Margulis measure
cites work: On the mixing property for hyperbolic systems
cites work: Weakly Differentiable Functions
cites work: Mixing, counting, and equidistribution in Lie groups
cites work: Some analytical properties of geodesically convex sets
cites work: On decay of correlations in Anosov flows
cites work: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups
cites work: The Patterson Measure for Geometrically Finite Groups with Parabolic Elements, New and Old
cites work: Spectra of Ruelle transfer operators for Axiom A flows
cites work: Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities
cites work: Shadow Lemma and nondivergence of the horospheres of a geometrically finite manifold.
cites work: Ergodicité et équidistribution en courbure négative
cites work: On the orbit function of discrete groups in negative curvature
cites work: Equidistribution, counting and approximation by quadratic irrationals
cites work: Variational principle and Kleinian groups
cites work: Equidistribution and counting for orbits of geometrically finite hyperbolic groups
cites work: The asymptotic distribution of circles in the orbits of Kleinian groups
cites work: The asymptotic distribution of Frobenius numbers
cites work: Certain measures associated with U-flows on compact manifolds
cites work: Applications of ergodic theory to the investigation of manifolds of negative curvature
cites work: On contact Anosov flows
cites work: Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds
cites work: Logarithm laws for flows on homogeneous spaces
DOI: 10.1017/ETDS.2012.192
Recommended article: Equidistribution of horospheres in non-positive curvature
Recommended article: Curvature properties of Riemannian manifolds with skew-circulant structures
Recommended article: On the mean curvature of submanifolds with nullity
Recommended article: Curvature-dimension inequalities and Li-Yau inequalities in sub-Riemannian spaces
Recommended article: Diameter estimates for submanifolds in manifolds with nonnegative curvature
Recommended article: On mean curvatures in submanifolds geometry
Recommended article: On the Mean Curvature Estimates for Bounded Submanifolds
Recommended article: Sublinear measures, Menger curvature, and Hausdorff dimension
Recommended article: A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality
Recommended article: ON THE DISTRIBUTION OF HYPERSURFACES EQUIDISTANT FROM TOTALLY GEODESIC SUBMANIFOLDS IN HYPERBOLIC SPACE
review text: 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…
reviewed by: Patrick Eberlein
description: scientific article; zbMATH DE number 6359413