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

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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

Abstract
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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.

Keywords: nonuniformly hyperbolic systems, periodic orbits., Geodesic flows.

Mathematics Subject Classification: Primary: 37C27; Secondary: 53D25, 37D2.

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:

[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).

show all references

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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