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Equidistribution results for geodesic flows

Published online by Cambridge University Press:  04 January 2013

ABDELHAMID AMROUN
ABDELHAMID AMROUN*
Affiliation:
Département de Mathématiques, Université Paris Sud, CNRS UMR 8628, 91405 Orsay Cedex, France (email: Abdelhamid.Amroun@math.u-psud.fr)
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Abstract

Using the works of Mañé [On the topological entropy of the geodesic flows. J. Differential Geom.45 (1989), 74–93] and Paternain [Topological pressure for geodesic flows. Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), 121–138] we study the distribution of geodesic arcs with respect to equilibrium states of the geodesic flow on a closed manifold, equipped with a $\mathcal {C}^{\infty }$Riemannian metric. We prove large-deviation lower and upper bounds and a contraction principle for the geodesic flow in the space of probability measures of the unit tangent bundle. We deduce a way of approximating equilibrium states for continuous potentials.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 3 , June 2014 , pp. 742 - 764
Copyright
©2013 Cambridge University Press 

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References

[1]Berger, M. and Bott, R.. Sur les variétés à courbure strictement positive. Topology 1 (1962).Google Scholar
[2]Burns, K. and Gutkin, E.. Growth of the number of geodesics between points and insecurity for Riemannian manifolds. Discrete Contin. Dyn. Syst. 21(2) (2008).Google Scholar
[3]Chengbo, Y.. Rigidity and dynamics around manifolds of negative curvature. Math. Res. Lett. 1 (1994), 123–147.Google Scholar
[4]Chengbo, Y.. The ergodic theory of discrete isometry groups on manifolds of variable negative curvature. Trans. Amer. Math. Soc. 348(12) (1996), 4965–5005.Google Scholar
[5]Dembo, A. and Zeitouni, O.. Large Deviations Techniques and Applications (Applications of Mathematics, 38). Springer, New York, 1998.Google Scholar
[6]Giles, J. R.. Convex Analysis with Application in Differentiation of Convex Functions (Research Notes in Mathematics, 58). Pitman, Boston, 1982.Google Scholar
[7]Keller, G.. Equilibrium States in Ergodic Theory (London Mathematical Society Study Texts, 42). Cambridge University Press, Cambridge, 1998.Google Scholar
[8]Kifer, Y.. Large deviations in dynamical systems and stochastic processes. Trans. Amer. Math. Soc. 321 (1988), 505–525.CrossRefGoogle Scholar
[9]Knieper, G.. The uniqueness of the measure of maximal entropy for geodesic flow on rank one manifolds. Ann. of Math. (2) 148 (1998), 291–314.Google Scholar
[10]Mañé, R.. On the topological entropy of the geodesic flows. J. Differential Geom. 45 (1989), 74–93.Google Scholar
[11]Newhouse, S.. Continuity properties of entropy. Ann. of Math. (2) 129 (1989), 215–235.Google Scholar
[12]Paternain, G. P.. Topological pressure for geodesic flows. Ann. Sci. Éc. Norm. Supér. (4) 33 (2000), 121–138.Google Scholar
[13]Paternain, G. P.. Geodesic Flows (Progress in Mathematics, 180). Birkhäuser, Basel, 2000.Google Scholar
[14]Rockafellar, R. T.. Convex Analysis. Princeton University Press, Princeton, NJ, 1970.Google Scholar
[15]Ruelle, D.. Thermodynamical Formalism (Encyclopedia of Mathematics and its Applications, 5). Addison-Wesley, Reading, MA, 1978.Google Scholar
[16]Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79). Springer, Berlin, 1982.Google Scholar