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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

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.  Google Scholar

[2]

L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, American Mathematical Society, Providence, R.I., 2002.  Google Scholar

[3]

M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, New York, 2002. doi: 10.1017/CBO9780511755316.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

N. Chernov and R. Markarian, Chaotic Billiards, American Mathematical Society, Providence, R.I., 2006. doi: 10.1090/surv/127.  Google Scholar

[8]

M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.  Google Scholar

[9]

M. P. Do Carmo, Riemannian Geometry, Birkhäuser Boston, Inc., Boston, MA, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

P. Eberlein, Geometry of Nonpositively Curved Manifolds, University of Chicago Press, Chicago, IL, 1996.  Google Scholar

[14]

D. Genin, Regular and Chaotic Dynamics of Outer Billiards, Ph.D. Thesis, Pennsylvania State University, 2005.  Google Scholar

[15]

R. Gunesch, Precise Asymptotics for Periodic Orbits of the Geodesic Flow in Nonpositive Curvature, Ph.D. Thesis, Pennsylvania State University, 2002.  Google Scholar

[16]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[17]

V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271. doi: 10.1007/s002200050811.  Google Scholar

[18]

A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2), 110 (1979), 529-547. doi: 10.2307/1971237.  Google Scholar

[19]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

G. Margulis, On Some Aspects of the Theory of Anosov Systems, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1.  Google Scholar

[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.  Google Scholar

[25]

S. Tabachnikov, Geometry and Billiards, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005.  Google Scholar

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.  Google Scholar

[2]

L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, American Mathematical Society, Providence, R.I., 2002.  Google Scholar

[3]

M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, New York, 2002. doi: 10.1017/CBO9780511755316.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

N. Chernov and R. Markarian, Chaotic Billiards, American Mathematical Society, Providence, R.I., 2006. doi: 10.1090/surv/127.  Google Scholar

[8]

M. P. Do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1976.  Google Scholar

[9]

M. P. Do Carmo, Riemannian Geometry, Birkhäuser Boston, Inc., Boston, MA, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

P. Eberlein, Geometry of Nonpositively Curved Manifolds, University of Chicago Press, Chicago, IL, 1996.  Google Scholar

[14]

D. Genin, Regular and Chaotic Dynamics of Outer Billiards, Ph.D. Thesis, Pennsylvania State University, 2005.  Google Scholar

[15]

R. Gunesch, Precise Asymptotics for Periodic Orbits of the Geodesic Flow in Nonpositive Curvature, Ph.D. Thesis, Pennsylvania State University, 2002.  Google Scholar

[16]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511809187.  Google Scholar

[17]

V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271. doi: 10.1007/s002200050811.  Google Scholar

[18]

A. Katok, Bernoulli diffeomorphisms on surfaces, Ann. of Math. (2), 110 (1979), 529-547. doi: 10.2307/1971237.  Google Scholar

[19]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137-173.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

G. Margulis, On Some Aspects of the Theory of Anosov Systems, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-662-09070-1.  Google Scholar

[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.  Google Scholar

[25]

S. Tabachnikov, Geometry and Billiards, American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005.  Google Scholar

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