Hyperbolic systems on nilpotent covers
Bulletin de la Société Mathématique de France, Volume 131 (2003) no. 2, pp. 267-287.

We study the ergodicity of the weak and strong stable foliations of hyperbolic systems on nilpotent covers. Subshifts of finite type and geodesic flows on negatively curved manifolds are also considered.

Nous étudions les propriétés ergodiques des feuilletages stables forts et faibles des systèmes hyperboliques définis sur un revêtement nilpotent. Les chaînes de Markov et les flots géodésiques en courbure négative sont aussi étudiés.

DOI: 10.24033/bsmf.2443
Classification: 37D10,  37D20,  37D40
Keywords: covering space, ergodic theory, geodesic flow, hyperbolic flow, invariant manifolds, Markov chain
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     title = {Hyperbolic systems on nilpotent covers},
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Coudene, Yves. Hyperbolic systems on nilpotent covers. Bulletin de la Société Mathématique de France, Volume 131 (2003) no. 2, pp. 267-287. doi : 10.24033/bsmf.2443. http://archive.numdam.org/articles/10.24033/bsmf.2443/

[1] J. Aaronson, R. Solomyak & O. Sarig - « Tail-invariant measures for some suspension semiflows », to appear. | MR | Zbl

[2] D. Anosov - Geodesic flows on closed riemannian manifolds with negative curvature, vol. 90, American Mathematical Society, Providence, R.I., 1967, English translation 1969. | MR | Zbl

[3] M. Babillot & F. Ledrappier - « Geodesic paths and horocycle flow on Abelian covers », Proceedings of the International Colloquium on Lie groups and ergodic theory (Mumbai 1996), Narosa Publishing House, New-Dehli, 1998, p. 1-32. | MR | Zbl

[4] B. Bowditch - « Geometrical finiteness with variable curvature », 77 (1995), no. 1, p. 229-274. | MR | Zbl

[5] R. Bowen - « Periodic orbits for hyperbolic flows », 94 (1972), p. 1-30. | MR | Zbl

[6] R. Bowen & B. Marcus - « Unique ergodicity for horocycle foliations », 13 (1977), p. 43-67. | MR | Zbl

[7] R. Bowen & D. Ruelle - « The ergodic theory of Axiom A flows », 29 (1975), p. 153-170. | MR | Zbl

[8] M. Brin - « Ergodicity of the geodesic flow », Notes from Mathematical Research Summer Institute, Seattle, 1999.

[9] Y. Coudene - « Gibbs measures on negatively curved manifolds », to appear. | MR | Zbl

[10] -, « Cocycles and stable foliations of axiom a flows », 21 (2001), p. 767-775. | MR | Zbl

[11] F. Dal'Bo - « Remarques sur le spectre des longueurs d'une surface et comptages », Bol. Soc. Brasil. Mat. (new series) 30 (1999), no. 2, p. 199-221. | MR | Zbl

[12] -, « Topologie du feuilletage fortement stable », 50 (2000), no. 3, p. 981-993. | Numdam | MR | Zbl

[13] F. Dal'Bo & M. Peigné - « Some negatively curved manifolds with cusps, mixing and counting », 497 (1998), p. 141-169. | MR | Zbl

[14] P. Eberlein - « Geodesic flows on negatively curved manifolds I », 95 (1972), p. 492-510. | MR | Zbl

[15] H. Furstenberg - « The unique ergodicity of the horocycle flow », Recent advances in topological dynamics, vol. 318, Springer, 1973, p. 95-115. | MR | Zbl

[16] U. Hamenstädt - « Ergodic properties of gibbs measures on nilpotent covers », to appear. | MR | Zbl

[17] G. Hedlund - « Fuchsian groups and transitive horocycles », 2 (1936), p. 530-542. | JFM | MR

[18] -, « Fuchsian groups and mixtures », 40 (1939), p. 370-383. | JFM | MR | Zbl

[19] E. Hopf - « Fuchsian groups and ergodic theory », 39 (1936), p. 299-314. | JFM | MR | Zbl

[20] V. Kaimanovich - « Ergodic properties of the horocycle flow and classification of Fuchsian groups », J. Dynam. Control Sys. 6 (2000), no. 1, p. 21-56. | MR | Zbl

[21] V. Kaimanovich & K. Schmidt - « Ergodicity of cocycles 1: general theory », to appear.

[22] A. Livsic - « Cohomology of dynamical systems », 6 (1972), p. 1278-1301. | MR | Zbl

[23] M. Pollicott - « d covers of horosphere foliations », Discrete Cont. Dynam. Sys. 6 (2000), no. 1, p. 147-154. | MR | Zbl

[24] D. Ruelle - Thermodynamical formalism, Encyclopedia of Mathematics and its applications, vol. 5, Addison-Wesley Publishing Compagny, Reading MA, 1978. | MR | Zbl

[25] M. Shub - Global stability of dynamical systems; with the collaboration of A.Fathi, R.Langevin, Springer Verlag, New York, etc., 1987, Transl. from French by J.Christy. | MR | Zbl

[26] S. Smale - « Differentiable dynamical systems », 73 (1967), p. 747-817. | MR | Zbl

[27] R. Solomyak - « A short proof of ergodicity of Babillot-Ledrappier measures », 129 (2001), no. 12, p. 3589-3591. | MR | Zbl

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