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

Bryce Weaver ^1,

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), 61-76. doi: 10.1007/BF02773153.
[2]	L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, American Mathematical Society, Providence, R.I., 2002.
[3]	M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, New York, 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), 1509-1527. doi: 10.1142/S0218127497001199.
[5]	K. Burns and M. Gerber, Real analytic Bernoulli geodesic flows on $S^2$, Ergod. Th. Dynam. Sys., 9 (1989), 27-45. 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), 757-785. doi: 10.1017/S0143385700008142.
[7]	N. Chernov and R. Markarian, Chaotic Billiards, American Mathematical Society, Providence, R.I., 2006. doi: 10.1090/surv/127.
[8]	M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.
[9]	M. P. Do Carmo, Riemannian Geometry, Birkhäuser Boston, Inc., Boston, MA, 1992.
[10]	V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy, Ergod. Th. Dynam. Sys., 8 (1988), 531-553. doi: 10.1017/S0143385700004685.
[11]	V. Donnay, Geodesic flow on the two-sphere. II. Ergodicity, in Dynamical Systems, Lecture Notes in Math., 1342, Springer, Berlin, 1988, 112-153. 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), 267-302. doi: 10.1007/BF02098044.
[13]	P. Eberlein, Geometry of Nonpositively Curved Manifolds, University of Chicago Press, Chicago, IL, 1996.
[14]	D. Genin, Regular and Chaotic Dynamics of Outer Billiards, Ph.D. Thesis, Pennsylvania State University, 2005.
[15]	R. Gunesch, Precise Asymptotics for Periodic Orbits of the Geodesic Flow in Nonpositive Curvature, Ph.D. Thesis, Pennsylvania State University, 2002.
[16]	B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.
[17]	V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271. doi: 10.1007/s002200050811.
[18]	A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2), 110 (1979), 529-547. doi: 10.2307/1971237.
[19]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.
[20]	A. Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, 1245-1253.
[21]	G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds, Ann. of Math. (2), 148 (1998), 291-314. doi: 10.2307/120995.
[22]	G. Knieper, Hyperbolic dynamics and Riemannian geometry, in Handbook of Dynamical Systems, Vol. 1A (eds. B. Hasselblatt and A. Katok), North-Holland, Amsterdam, 2002, 453-545. doi: 10.1016/S1874-575X(02)80008-X.
[23]	G. Margulis, On Some Aspects of the Theory of Anosov Systems, Springer-Verlag, Berlin, 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), 573-591. doi: 10.2307/2006982.
[25]	S. Tabachnikov, Geometry and Billiards, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005.

show all references

References:

[1]	M. Babillot, On the mixing property for hyperbolic systems, Israel J. Math., 129 (2002), 61-76. doi: 10.1007/BF02773153.
[2]	L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, American Mathematical Society, Providence, R.I., 2002.
[3]	M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, New York, 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), 1509-1527. doi: 10.1142/S0218127497001199.
[5]	K. Burns and M. Gerber, Real analytic Bernoulli geodesic flows on $S^2$, Ergod. Th. Dynam. Sys., 9 (1989), 27-45. 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), 757-785. doi: 10.1017/S0143385700008142.
[7]	N. Chernov and R. Markarian, Chaotic Billiards, American Mathematical Society, Providence, R.I., 2006. doi: 10.1090/surv/127.
[8]	M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.
[9]	M. P. Do Carmo, Riemannian Geometry, Birkhäuser Boston, Inc., Boston, MA, 1992.
[10]	V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy, Ergod. Th. Dynam. Sys., 8 (1988), 531-553. doi: 10.1017/S0143385700004685.
[11]	V. Donnay, Geodesic flow on the two-sphere. II. Ergodicity, in Dynamical Systems, Lecture Notes in Math., 1342, Springer, Berlin, 1988, 112-153. 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), 267-302. doi: 10.1007/BF02098044.
[13]	P. Eberlein, Geometry of Nonpositively Curved Manifolds, University of Chicago Press, Chicago, IL, 1996.
[14]	D. Genin, Regular and Chaotic Dynamics of Outer Billiards, Ph.D. Thesis, Pennsylvania State University, 2005.
[15]	R. Gunesch, Precise Asymptotics for Periodic Orbits of the Geodesic Flow in Nonpositive Curvature, Ph.D. Thesis, Pennsylvania State University, 2002.
[16]	B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.
[17]	V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271. doi: 10.1007/s002200050811.
[18]	A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2), 110 (1979), 529-547. doi: 10.2307/1971237.
[19]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.
[20]	A. Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, 1245-1253.
[21]	G. Knieper, The uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds, Ann. of Math. (2), 148 (1998), 291-314. doi: 10.2307/120995.
[22]	G. Knieper, Hyperbolic dynamics and Riemannian geometry, in Handbook of Dynamical Systems, Vol. 1A (eds. B. Hasselblatt and A. Katok), North-Holland, Amsterdam, 2002, 453-545. doi: 10.1016/S1874-575X(02)80008-X.
[23]	G. Margulis, On Some Aspects of the Theory of Anosov Systems, Springer-Verlag, Berlin, 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), 573-591. doi: 10.2307/2006982.
[25]	S. Tabachnikov, Geometry and Billiards, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005.

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