Growth of periodic orbits and generalized diagonals for typical triangular billiards

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January 2013, 7(1): 31-44. doi: 10.3934/jmd.2013.7.31

Growth of periodic orbits and generalized diagonals for typical triangular billiards

Dmitri Scheglov ^1,

1.	Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, United States

Received May 2012 Revised October 2012 Published May 2013

Abstract
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We prove that for any $\epsilon>0$ the growth rate $P_n$ of generalized diagonals or periodic orbits of a typical (in the Lebesgue measure sense) triangular billiard satisfies: $P_n < Ce^{n^{\sqrt{3}-1+\epsilon}}$. This provides an explicit subexponential estimate on the triangular billiard complexity and answers a long-standing open question for typical triangles. This also makes progress towards a solution of Problem 3 in Katok's list of "Five most resistant problems in dynamics". The proof uses essentially new geometric ideas and does not rely on the rational approximations.

Keywords: Billiards, complexity..

Mathematics Subject Classification: Primary: 37C10; Secondary: 37C3.

Citation: Dmitri Scheglov. Growth of periodic orbits and generalized diagonals for typical triangular billiards. Journal of Modern Dynamics, 2013, 7 (1) : 31-44. doi: 10.3934/jmd.2013.7.31

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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