ebsd2021:knieper []

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====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, [[https://link.springer.com/book/10.1007/978-3-662-12494-9|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, [[https://arxiv.org/abs/1903.09831|arXiv:1903.09831]] Adv. Math., 376, (2021), 44 pp.

[8] V. Climenhaga, G. Knieper , K. War, Closed geodesics on surfaces
without conjugate points, [[https://arxiv.org/abs/2008.02249|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, [[https://link.springer.com/book/10.1007/978-3-662-09070-1|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])

[[http://www.im.ufrj.br/dynsys/2021-EBSD/seminario-1-Alexander.pdf|Introdução, subespaços vertical e horizontal, campos de Jacobi, definição do fluxo geodésico, forma simplética (Alexander Cantoral)]]

[[http://www.im.ufrj.br/dynsys/2021-EBSD/seminar-2-Nestor.pdf|Equação de Jacobi, wronskiano, Lagrangianos e relação com pontos conjugados (Nestor)]]

[[http://www.im.ufrj.br/dynsys/2021-EBSD/seminar-3-Nestor.pdf|Lema do índice, teorema de Rauch, estimativas de hiperbolicidade em curvatura negativa
 (Nestor)]]

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[[ebsd2021:tema1|The geodesic flow, the tangent bundle $TTM$, Jacobi fields
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[[ebsd2021:tema2|Geodesic flows in Hamiltonian dynamics
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[[ebsd2021:tema3|Jacobi equation in Hamiltonian dynamics
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[[ebsd2021:tema4|Tema 4 
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[[ebsd2021:tema5|Tema 5
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[[ebsd2021:tema6|Hadamard manifolds - part 1
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[[ebsd2021:tema7|Hadarmard manifolds - part 2
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[[ebsd2021:tema8|Boundary at infinity of Hadarmard manifolds
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[[ebsd2021:tema9|Busemann functions and horospherical foliations
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[[ebsd2021:tema10|Poincaré series
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[[ebsd2021:tema11|Limit set
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[[ebsd2021:tema12|Busemann density
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