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Growth of periodic orbits and generalized diagonals for typical triangular billiards
1. | Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, United States |
References:
[1] |
J. Cassaigne, P. Hubert and S. Troubetzkoy, Complexity and growth for polygonal billiards, Ann. Inst. Fourier (Grenoble), 52 (2002), 835-847. |
[2] |
E. Gutkin and M. Rams, Growth rates for geometric complexities and counting functions in polygonal billiards, Ergodic Theory Dynam. Systems, 29 (2009), 1163-1183.
doi: 10.1017/S0143385708080620. |
[3] |
E. Gutkin and S. Tabachnikov, Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards on surfaces of constant curvature, (English summary) Mosc. Math. J., 6 (2006), 673-701, 772. |
[4] |
E. Gutkin and S. Troubetzkoy, Directional flows and strong recurrence for polygonal billiards,, Proceedings of the International Congress of Dynamical Systems, ().
|
[5] |
B. Hasselblatt, ed., "Dynamics, Ergodic Theory, and Geometry," Mathematical Sciences Research Institute Publications, 54, Cambridge University Press, Cambridge, 2007.
doi: 10.1017/CBO9780511755187. |
[6] |
V. Kaloshin and I. Rodnianski, Diophantine properties of elements of SO(3), Geom. Funct. Anal., 11 (2001), 953-970.
doi: 10.1007/s00039-001-8222-8. |
[7] |
A. Katok, Five most resistant problems in dynamics., Available from: \url{http://www.math.psu.edu/katok_a/pub/5problems-expanded.pdf}., ().
|
[8] |
A. Katok, The growth rate for the number of singular and periodic orbits for a polygonal billiard, Comm. Math. Phys., 111 (1987), 151-160.
doi: 10.1007/BF01239021. |
[9] |
A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons, Mat. Zametki, 18 (1975), 291-300. |
[10] |
H. Masur, The growth rate of trajectories of a quadratic differential, Ergod. Th. Dyn. Sys., 10 (1990), 151-176.
doi: 10.1017/S0143385700005459. |
[11] |
H. Masur, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential, in "Holomorphic Functions and Moduli, Vol. 1" (ed. D. Drasin) (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988.
doi: 10.1007/978-1-4613-9602-4_20. |
[12] |
D. Scheglov, Lower bounds on directional complexity for irrational triangle billiards,, preprint., ().
|
[13] |
S. Troubetzkoy, Complexity lower bounds for polygonal billiards, Chaos, 8 (1998), 242-244.
doi: 10.1063/1.166301. |
[14] |
Y. Vorobets, Ergodicity of billiards in polygons, Mat. Sb., 188 (1997), 65-112.
doi: 10.1070/SM1997v188n03ABEH000211. |
show all references
References:
[1] |
J. Cassaigne, P. Hubert and S. Troubetzkoy, Complexity and growth for polygonal billiards, Ann. Inst. Fourier (Grenoble), 52 (2002), 835-847. |
[2] |
E. Gutkin and M. Rams, Growth rates for geometric complexities and counting functions in polygonal billiards, Ergodic Theory Dynam. Systems, 29 (2009), 1163-1183.
doi: 10.1017/S0143385708080620. |
[3] |
E. Gutkin and S. Tabachnikov, Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards on surfaces of constant curvature, (English summary) Mosc. Math. J., 6 (2006), 673-701, 772. |
[4] |
E. Gutkin and S. Troubetzkoy, Directional flows and strong recurrence for polygonal billiards,, Proceedings of the International Congress of Dynamical Systems, ().
|
[5] |
B. Hasselblatt, ed., "Dynamics, Ergodic Theory, and Geometry," Mathematical Sciences Research Institute Publications, 54, Cambridge University Press, Cambridge, 2007.
doi: 10.1017/CBO9780511755187. |
[6] |
V. Kaloshin and I. Rodnianski, Diophantine properties of elements of SO(3), Geom. Funct. Anal., 11 (2001), 953-970.
doi: 10.1007/s00039-001-8222-8. |
[7] |
A. Katok, Five most resistant problems in dynamics., Available from: \url{http://www.math.psu.edu/katok_a/pub/5problems-expanded.pdf}., ().
|
[8] |
A. Katok, The growth rate for the number of singular and periodic orbits for a polygonal billiard, Comm. Math. Phys., 111 (1987), 151-160.
doi: 10.1007/BF01239021. |
[9] |
A. Katok and A. Zemlyakov, Topological transitivity of billiards in polygons, Mat. Zametki, 18 (1975), 291-300. |
[10] |
H. Masur, The growth rate of trajectories of a quadratic differential, Ergod. Th. Dyn. Sys., 10 (1990), 151-176.
doi: 10.1017/S0143385700005459. |
[11] |
H. Masur, Lower bounds for the number of saddle connections and closed trajectories of a quadratic differential, in "Holomorphic Functions and Moduli, Vol. 1" (ed. D. Drasin) (Berkeley, CA, 1986), Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988.
doi: 10.1007/978-1-4613-9602-4_20. |
[12] |
D. Scheglov, Lower bounds on directional complexity for irrational triangle billiards,, preprint., ().
|
[13] |
S. Troubetzkoy, Complexity lower bounds for polygonal billiards, Chaos, 8 (1998), 242-244.
doi: 10.1063/1.166301. |
[14] |
Y. Vorobets, Ergodicity of billiards in polygons, Mat. Sb., 188 (1997), 65-112.
doi: 10.1070/SM1997v188n03ABEH000211. |
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