Abstract
We consider the geodesic flow on a complete connected negatively curved manifold. We show that the set of invariant borel probability measures contains a dense G δ -subset consisting of ergodic measures fully supported on the non-wandering set. We also treat the case of non-positively curved manifolds and provide general tools to deal with hyperbolic systems defined on non-compact spaces.
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References
D. V. Anosov, Geodesic flows on closed riemannian manifolds with negative curvature, Proceedings of the Steklov Institute of Mathematics 90 (1967).
W. Ballmann, Lectures on Spaces of Nonpositive Curvature, With an appendix by Misha Brin, DMV Seminar, Vol. 25, Birkhauser Verlag, Basel, 1995.
W. Ballmann, M. Brin and P. Eberlein, Structure of manifolds of nonpositive curvature. I, Annals of Mathematics. Second Series 122 (1985), 171–203.
P. Billingsley, Convergence of Probability Measures, 2nd edn., Wiley Series in Probability and Statistics, Wiley, Chichester, 1999, x+277 pp.
Y. Coudene, Une version mesurable du théorème de Stone-Weierstrass, Gazette des mathematiciens 91 (2002), 10–17.
Y. Coudene, Gibbs measures on negatively curved manifolds, Journal of Dynamical and Control System 9 (2003), 89–101.
Y. Coudene, Topological dynamics and local product structure, Journal of the London Mathematical Society. Second Series 69 (2004), 441–456.
P. Eberlein, Geodesic flows on negatively curved manifolds I., Annals of Mathematics. Second Series 95 (1972), 492–510.
P. Eberlein, Geometry of Nonpositively Curved Manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1996, vii+449 pp.
P. Eberlein and B. O’Neill, Visibility manifolds, Pacific Journal of Mathematics 46 (1973), 45–109.
Y. Guivarc’h, Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés, Ergodic Theory and Dynamical Systems 9 (1989), 433–453.
E. Hopf, Fuchsian groups and ergodic theory, Transactions of the American Mathematical Society 39 (1936), 299–314.
G. Knieper, Hyperbolic dynamics and Riemannian geometry, in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 453–545.
A. Livsic, Certain properties of the homology of Y -systems. (Russian) Rossiĭskaya Akademiya Nauk 10 (1971), 555–564. English translation: Math. Notes 10 (1971), 758–763.
J. C. Oxtoby, On two theorems of Parthasarathy and Kakutani concerning the shift transformation, in Ergodic Theory (Proc. Internat. Sympos., Tulane Univ., New Orleans, La., 1961) Academic Press, New York, 1963, pp. 203–215.
K. Sigmund, Generic properties of invariant measures for AxiomA-Diffeomorphisms, Inventiones Mathematicae 11 (1970), 99–109.
K. Sigmund, On the space of invariant measures for hyperbolic flows, American Journal of Mathematics 94 (1972), 31–37.
R. J. Zimmer, Amenable ergodic group actions and an application to Poisson boundaries of random walks, Functional Analysis 27 (1978), 350–372.
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Coudene, Y., Schapira, B. Generic measures for hyperbolic flows on non-compact spaces. Isr. J. Math. 179, 157–172 (2010). https://doi.org/10.1007/s11856-010-0076-z
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DOI: https://doi.org/10.1007/s11856-010-0076-z