Measure of maximal entropy and Margulis estimate on the number of periodic orbits in the framework of manifolds (in particular surfaces) without conjugate points.
One of the main questions in smooth ergodic theory is to which extend existence and uniqueness of equilibrium states carry over to dynamical systems with some coarse hyperbolicity. In this lectures, we will study these questions on a class of geometrically defined dynamical systems, namely geodesic flows on closed manifolds without conjugate points. Furthermore, we will assume the existence of a background metric of strictly negative curvature which will guarantee coarse hyperbolicity of our dynamical system. Unlike non-positive curvature the more general condition of no conjugate points is not a local assumption and certain rigidity properties like the flat strip theorem valid in non-positive curvature do not hold anymore. Nevertheless, we will show the existence of a measure of maximal entropy (MME). For surfaces we are able to prove mixing, uniqueness of the MME and the uniform distribution of closed orbits w.r.t. this measure. Using the mixing properties of the MME we are also able to obtain a precise asymptotics on the growth rate of closed geodesics. Due to the work of Margulis and Bowen this was previously known only in the case of strictly negative curvature (see [13], [14], [2] and [3]). In the case of non-positive curvature weaker estimates where obtained in [10] and [11] (see also [6] for new developments). We also formulate conditions for which these results carry over to higher dimensions. These lectures are to a large extend based on joint work with Vaughn Climenhaga and Khadim War (see [7],[8]).
Prerequisites:
1. Basics in Riemannian Geometry: Riemannian metrics, geodesics and geodesic flows, exponential map, manifolds of no conjugate points, Theorem of Hadamard Cartan, Busemann functions, special properties of manifolds with non-positive curvature and negative curvature, group of deck transformations and fundamental group, deck transformations and closed geodesics, Theorem of Preismann (see [9], [15], [1], [12]).
2. Gromov hyperbolicity: Quasi geodesics, Morse correspondence, Visibility boundary of a hyperbolic space (see e.g. [4], [12]).
3. Basic ergodic theory: ergodicity and mixing, measure of maximal entropy and variational principle, topological entropy (see e.g. [16]).
References
[1] W. Ballmann, M. Gromov, V. Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, 61, Birkhäuser Boston, Inc., Boston, MA, 1985.
[2] R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., American Journal of Mathematics, 94, 1972,1–30.
[3] R. Bowen, Maximizing entropy for a hyperbolic flow, Math. Systems Theory, 7, 1974.
[4] M. R. Bridson, A.Haefliger, Metric spaces of non-positive curvature, Fundamental Principles of Mathematical Sciences, Springer-Verlag, Berlin, 1999.
[5] M. Coornaert, G. Knieper, Growth of conjugacy classes in Gromov hyperbolic groups, Geom. Funct. Anal, 12 (2002), 464–478.
[6] K. Burns, K, V. Climenhaga, T. Fisher, D. Thompson, Unique equilibrium states for geodesic flows in nonpositive curvature, Geom. Funct. Anal. 28, 2018, 1209–1259.
[7] V. Climenhaga, G. Knieper , K. War, Uniqueness of the measure of maximal entropy for geodesic flows on certain manifolds without conjugate points, arXiv:1903.09831 Adv. Math., 376, (2021), 44 pp.
[8] V. Climenhaga, G. Knieper , K. War, Closed geodesics on surfaces without conjugate points, arXiv:2008.02249, to appear in Commun. Contemp. Math.
[9] M. do Carmo, Riemannian geometry, Birkhäuser Boston, Inc., Boston, MA, 1992.
[10] G. Knieper, On the asymptotic geometry of nonpositively curved manifolds, Geom. Funct. Anal, 7 (1997), 755–782.
[11] G. Knieper, The uniqueness of the measure of maximal entropy for rank 1 manifolds, Annals of Math. 148 (1998), 291–314.
[12] G. Knieper, Hyperbolic Dynamics and Riemannian Geometry, in Handbook of Dynamical Systems, Elsevier Science B., vol. 1A, (2002), 453–545.
[13] G. A. Margulis, Certain applications of ergodic theory to the investi gation of manifolds of negative
[14] G. A. Margulis, On some aspects of the theory of Anosov systems, With a survey by Richard Sharp: Periodic orbits of hyperbolic flows, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2004.
[15] S. Sakai, Riemannian geometry, Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1996.
[16] P. Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, 79 , Springer-Verlag, New York-Berlin, 1982.
==== Seminar talks ==== (seguindo [12])
Equação de Jacobi, wronskiano, Lagrangianos e relação com pontos conjugados (Nestor)
Lema do índice, teorema de Rauch, estimativas de hiperbolicidade em curvatura negativa (Nestor)