\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 37C27; Secondary: 53D25, 37D25.

    Citation:

    \begin{equation} \\ \end{equation}
  • [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.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(108) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return