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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 |
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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