No CrossRef data available.
Article contents
A dynamical-geometric characterization of the geodesic flows of negatively curved locally symmetric spaces
Published online by Cambridge University Press: 03 July 2014
Abstract
In this paper we prove the following rigidity result: let 
${\it\varphi}$ be a 
$C^{\infty }$ topologically mixing transversely symplectic Anosov flow. If (i) its weak stable and weak unstable distributions are 
$C^{\infty }$ and (ii) its Hamenstädt metrics are sub-Riemannian, then up to finite covers and a constant change of time scale, 
${\it\varphi}$ is 
$C^{\infty }$ flow conjugate to the geodesic flow of a closed locally symmetric Riemannian space of rank one.
- Type
- Research Article
- Information
- Copyright
- © Cambridge University Press, 2014
References
Anosov, V. D.. Geodesic flows on closed Riemannian manifolds with negative curvature. Proc. Inst. Steklov 90 (1967), 1–235.Google Scholar
Benoist, Y.. Orbites des structures rigides (d’après M. Gromov). Progr. Mat. 145 (1997), 1–17.Google Scholar
Benoist, Y.. Réseaux des groupes de Lie. Cours de M2 Ã. l’Université Paris 6, Paris, 2008.Google Scholar
Benoist, Y., Foulon, P. and Labourie, F.. Flots d’Anosov à distributions stable et instable différentiables. J. Amer. Math. Soc. 5 (1992), 33–74.Google Scholar
Candel, A. and Quiroga-Barranco, R.. Gromovs centralizer theorem. Geom. Dedicata 100 (2003), 123–155.CrossRefGoogle Scholar
D’ambra, G. and Gromov, M.. Lectures on transformation groups: geometry and dynamics. Surv. Differ. Geom. 1 (1991), 19–111.CrossRefGoogle Scholar
Fang, Y.. Smooth rigidity of uniformly quasiconformal Anosov flows. Ergod. Th. & Dynam. Sys. 24 (2004), 1937–1959.CrossRefGoogle Scholar
Fang, Y.. Invariant rigid geometric structures and expanding maps. Ergod. Th. & Dynam. Sys. 32 (2012), 941–959.CrossRefGoogle Scholar
Fang, Y.. Quasiconformal Anosov flows and quasisymmetric rigidity of Hamenstädt distances. Discrete Contin. Dyn. Syst. 34 (2014), 3471–3483.CrossRefGoogle Scholar
Fang, Y.. Structures géométriques rigides et systèmes dynamiques hyperboliques. PhD Thesis, Université de Paris-Sud, 2004.Google Scholar
Gromov, M.. Rigid transformation groups. Géométrie différentielle (Paris, 1986) (Travaux en Cours, 33). Eds. Bernard, D. and Choquet-Bruhat, Y.. Hermann, Paris, 1988, pp. 65–139.Google Scholar
Hamenstädt, U.. A new description of the Bowen–Margulis measure. Ergod. Th. & Dynam. Sys. 9 (1989), 455–464.CrossRefGoogle Scholar
Hamenstädt, U.. Entropy–rigidity of locally symmetric spaces of negative curvature. Ann. of Math. (2) 131 (1990), 35–51.CrossRefGoogle Scholar
Hamenstädt, U.. A geometric characterization of negatively curved locally symmetric spaces. J. Differ. Geom. 34 (1991), 193–221.CrossRefGoogle Scholar
Hasselblatt, B.. A new construction of the Margulis measure for Anosov flows. Ergod. Th. & Dynam. Sys. 9 (1989), 465–468.CrossRefGoogle Scholar
Hasselblatt, B. and Katok, A.. Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications, 54). Cambridge University Press, Cambridge, 1995.Google Scholar
Margulis, G. A.. Discrete Subgroups of Semisimple Lie Groups (Ergebnisse der Mathematick und ihrer Grenzgebiete, 17). Springer, Berlin, 1991.CrossRefGoogle Scholar
Mitchell, J.. On Carnot–Carathéodory metrics. J. Differ. Geom. 21 (1985), 35–45.CrossRefGoogle Scholar
Morris, D. W.. Ratner’s Theorems on Unipotent Flows. University of Chicago Press, Chicago, IL, 2005.Google Scholar
Tomter, P.. Anosov flows in infra-homogeneous spaces. Proc. Sympos. Pure Math. 14 (1970), 299–327.CrossRefGoogle Scholar
Tomter, P.. On the classification of Anosov flows. Topology 14 (1975), 179–189.CrossRefGoogle Scholar