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:
[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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