American Institute of Mathematical Sciences

Advanced
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.  doi: 10.1007/BF02773153.  Google Scholar

[2]

L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory,, American Mathematical Society, (2002).   Google Scholar

[3]

M. Brin and G. Stuck, Introduction to Dynamical Systems,, Cambridge University Press, (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.  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.  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.  doi: 10.1017/S0143385700008142.  Google Scholar

[7]

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

[8]

M. P. Do Carmo, Differential Geometry of Curves and Surfaces,, Prentice-Hall, (1976).   Google Scholar

[9]

M. P. Do Carmo, Riemannian Geometry,, Birkhäuser Boston, (1992).   Google Scholar

[10]

V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy,, Ergod. Th. Dynam. Sys., 8 (1988), 531.  doi: 10.1017/S0143385700004685.  Google Scholar

[11]

V. Donnay, Geodesic flow on the two-sphere. II. Ergodicity,, in Dynamical Systems, (1342), 112.  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.  doi: 10.1007/BF02098044.  Google Scholar

[13]

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

[14]

D. Genin, Regular and Chaotic Dynamics of Outer Billiards,, Ph.D. Thesis, (2005).   Google Scholar

[15]

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

[16]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (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.  doi: 10.1007/s002200050811.  Google Scholar

[18]

A. Katok, Bernoulli diffeomorphisms on surfaces,, Ann. of Math. (2), 110 (1979), 529.  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.   Google Scholar

[20]

A. Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems,, in Proceedings of the International Congress of Mathematicians, (1983), 1245.   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.  doi: 10.2307/120995.  Google Scholar

[22]

G. Knieper, Hyperbolic dynamics and Riemannian geometry,, in Handbook of Dynamical Systems, (2002), 453.  doi: 10.1016/S1874-575X(02)80008-X.  Google Scholar

[23]

G. Margulis, On Some Aspects of the Theory of Anosov Systems,, Springer-Verlag, (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.  doi: 10.2307/2006982.  Google Scholar

[25]

S. Tabachnikov, Geometry and Billiards,, American Mathematical Society, (2005).   Google Scholar

show all references

References:
[1]

M. Babillot, On the mixing property for hyperbolic systems,, Israel J. Math., 129 (2002), 61.  doi: 10.1007/BF02773153.  Google Scholar

[2]

L. Barreira and Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory,, American Mathematical Society, (2002).   Google Scholar

[3]

M. Brin and G. Stuck, Introduction to Dynamical Systems,, Cambridge University Press, (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.  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.  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.  doi: 10.1017/S0143385700008142.  Google Scholar

[7]

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

[8]

M. P. Do Carmo, Differential Geometry of Curves and Surfaces,, Prentice-Hall, (1976).   Google Scholar

[9]

M. P. Do Carmo, Riemannian Geometry,, Birkhäuser Boston, (1992).   Google Scholar

[10]

V. Donnay, Geodesic flow on the two-sphere. I. Positive measure entropy,, Ergod. Th. Dynam. Sys., 8 (1988), 531.  doi: 10.1017/S0143385700004685.  Google Scholar

[11]

V. Donnay, Geodesic flow on the two-sphere. II. Ergodicity,, in Dynamical Systems, (1342), 112.  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.  doi: 10.1007/BF02098044.  Google Scholar

[13]

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

[14]

D. Genin, Regular and Chaotic Dynamics of Outer Billiards,, Ph.D. Thesis, (2005).   Google Scholar

[15]

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

[16]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems,, Cambridge University Press, (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.  doi: 10.1007/s002200050811.  Google Scholar

[18]

A. Katok, Bernoulli diffeomorphisms on surfaces,, Ann. of Math. (2), 110 (1979), 529.  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.   Google Scholar

[20]

A. Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems,, in Proceedings of the International Congress of Mathematicians, (1983), 1245.   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.  doi: 10.2307/120995.  Google Scholar

[22]

G. Knieper, Hyperbolic dynamics and Riemannian geometry,, in Handbook of Dynamical Systems, (2002), 453.  doi: 10.1016/S1874-575X(02)80008-X.  Google Scholar

[23]

G. Margulis, On Some Aspects of the Theory of Anosov Systems,, Springer-Verlag, (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.  doi: 10.2307/2006982.  Google Scholar

[25]

S. Tabachnikov, Geometry and Billiards,, American Mathematical Society, (2005).   Google Scholar

[1]

Carlos Arnoldo Morales. A note on periodic orbits for singular-hyperbolic flows. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 615-619. doi: 10.3934/dcds.2004.11.615
[2]

Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505
[3]

Zheng Yin, Ercai Chen. Conditional variational principle for the irregular set in some nonuniformly hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6581-6597. doi: 10.3934/dcds.2016085
[4]

Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022
[5]

Alain Jacquemard, Weber Flávio Pereira. On periodic orbits of polynomial relay systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 331-347. doi: 10.3934/dcds.2007.17.331
[6]

César J. Niche. Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles. Discrete & Continuous Dynamical Systems - A, 2006, 14 (4) : 617-630. doi: 10.3934/dcds.2006.14.617
[7]

Răzvan M. Tudoran. On the control of stability of periodic orbits of completely integrable systems. Journal of Geometric Mechanics, 2015, 7 (1) : 109-124. doi: 10.3934/jgm.2015.7.109
[8]

Flaviano Battelli, Ken Palmer. Transversal periodic-to-periodic homoclinic orbits in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 367-387. doi: 10.3934/dcdsb.2010.14.367
[9]

Luis Barreira, Claudia Valls. Existence of stable manifolds for nonuniformly hyperbolic $c^1$ dynamics. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 307-327. doi: 10.3934/dcds.2006.16.307
[10]

Victoriano Carmona, Emilio Freire, Soledad Fernández-García. Periodic orbits and invariant cones in three-dimensional piecewise linear systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 59-72. doi: 10.3934/dcds.2015.35.59
[11]

Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807
[12]

B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217
[13]

Gabriele Benedetti, Kai Zehmisch. On the existence of periodic orbits for magnetic systems on the two-sphere. Journal of Modern Dynamics, 2015, 9: 141-146. doi: 10.3934/jmd.2015.9.141
[14]

Armengol Gasull, Héctor Giacomini, Maite Grau. On the stability of periodic orbits for differential systems in $\mathbb{R}^n$. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 495-509. doi: 10.3934/dcdsb.2008.10.495
[15]

Armengol Gasull, Víctor Mañosa. Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 651-670. doi: 10.3934/dcdsb.2019259
[16]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[17]

Daniel Visscher. A new proof of Franks' lemma for geodesic flows. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4875-4895. doi: 10.3934/dcds.2014.34.4875
[18]

Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841
[19]

Anatole Katok, Federico Rodriguez Hertz. Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups. Journal of Modern Dynamics, 2010, 4 (3) : 487-515. doi: 10.3934/jmd.2010.4.487
[20]

Weisheng Wu. Modified Schmidt games and non-dense forward orbits of partially hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3463-3481. doi: 10.3934/dcds.2016.36.3463

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences