# American Institute of Mathematical Sciences

April  2014, 8(2): 139-176. doi: 10.3934/jmd.2014.8.139

## Growth rate of periodic orbits for geodesic flows over surfaces with radially symmetric focusing caps

 1 Department of Mathematics, Indiana University, Rawles Hall, 831 East 3rd St, Bloomington, IN 47405, United States

Received  November 2011 Revised  August 2014 Published  November 2014

We obtain a precise asymptotic formula for the growth rate of periodic orbits of the geodesic flow over metrics on surfaces with negative curvature outside of a disjoint union of radially symmetric focusing caps of positive curvature. This extends results of G. Margulis and G. Knieper for negative and nonpositive curvature respectively.
Citation: Bryce Weaver. Growth rate of periodic orbits for geodesic flows over surfaces with radially symmetric focusing caps. Journal of Modern Dynamics, 2014, 8 (2) : 139-176. doi: 10.3934/jmd.2014.8.139
##### References:
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##### References:
 [1] M. Babillot, On the mixing property for hyperbolic systems,, Israel J. Math., 129 (2002), 61. doi: 10.1007/BF02773153. [2] L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory,, American Mathematical Society, (2002). [3] M. Brin and G. Stuck, Introduction to Dynamical Systems,, Cambridge University Press, (2002). doi: 10.1017/CBO9780511755316. [4] K. Burns and V. Donnay, Embedded surfaces with ergodic geodesic flow,, Inter. J. Bifur. Chaos Appl. Sci. Engrg., 7 (1997), 1509. doi: 10.1142/S0218127497001199. [5] K. Burns and M. Gerber, Real analytic Bernoulli geodesic flows on $S^2$,, Ergod. Th. Dynam. Sys., 9 (1989), 27. doi: 10.1017/S0143385700004806. [6] K. Burns and A. Katok, Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dynamical systems,, Erg. Theory Dynam. Systems, 14 (1994), 757. doi: 10.1017/S0143385700008142. [7] N. Chernov and R. Markarian, Chaotic Billiards,, American Mathematical Society, (2006). doi: 10.1090/surv/127. [8] M. P. Do Carmo, Differential Geometry of Curves and Surfaces,, Prentice-Hall, (1976). [9] M. P. Do Carmo, Riemannian Geometry,, Birkhäuser Boston, (1992). [10] V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy,, Ergod. Th. Dynam. Sys., 8 (1988), 531. doi: 10.1017/S0143385700004685. [11] V. Donnay, Geodesic flow on the two-sphere. II. Ergodicity,, in Dynamical Systems, (1342), 112. doi: 10.1007/BFb0082827. [12] V. Donnay and C. Liverani, Potentials on the two-torus for which the Hamiltonian flow is ergodic,, Commun. Math. Phys., 135 (1991), 267. doi: 10.1007/BF02098044. [13] P. Eberlein, Geometry of Nonpositively Curved Manifolds,, University of Chicago Press, (1996). [14] D. Genin, Regular and Chaotic Dynamics of Outer Billiards,, Ph.D. Thesis, (2005). [15] R. Gunesch, Precise Asymptotics for Periodic Orbits of the Geodesic Flow in Nonpositive Curvature,, Ph.D. Thesis, (2002). [16] B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511809187. [17] V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits,, Comm. Math. Phys., 211 (2000), 253. doi: 10.1007/s002200050811. [18] A. Katok, Bernoulli diffeomorphisms on surfaces,, Ann. of Math. (2), 110 (1979), 529. doi: 10.2307/1971237. [19] A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137. [20] A. Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems,, in Proceedings of the International Congress of Mathematicians, (1983), 1245. [21] G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds,, Ann. of Math. (2), 148 (1998), 291. doi: 10.2307/120995. [22] G. Knieper, Hyperbolic dynamics and Riemannian geometry,, in Handbook of Dynamical Systems, (2002), 453. doi: 10.1016/S1874-575X(02)80008-X. [23] G. Margulis, On Some Aspects of the Theory of Anosov Systems,, Springer-Verlag, (2004). doi: 10.1007/978-3-662-09070-1. [24] W. Parry and M. Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flow,, Ann. of Math. (2), 118 (1983), 573. doi: 10.2307/2006982. [25] S. Tabachnikov, Geometry and Billiards,, American Mathematical Society, (2005).
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