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Growth rates for geometric complexities and counting functions in polygonal billiards

Published online by Cambridge University Press:  01 August 2009

EUGENE GUTKIN  and
MICHAŁ RAMS
EUGENE GUTKIN
Affiliation:
IMPA, Estrada Dona Castorina 110, Rio de Janeiro, 22460-320, Brazil UMK, Chopina 12/18, Torun 87-100, Poland (email: [email protected], [email protected])
MICHAŁ RAMS
Affiliation:
IMPAN, S̀niadeckich 8, 00-956 Warszawa 10, Poland (email: [email protected])
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Abstract

We introduce a new method for estimating the growth of various quantities arising in dynamical systems. Applying our method to polygonal billiards on surfaces of constant curvature, we obtain estimates almost everywhere on direction complexities and position complexities. As a byproduct, we recover the power bounds of Boshernitzan on the number of billiard orbits between almost all pairs of points in a planar polygon [M. Boshernitzan. Private letter to H. Masur (1986)].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 29 , Issue 4 , August 2009 , pp. 1163 - 1183
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Boshernitzan, M.. Private letter to H. Masur (1986).Google Scholar
[2]Davis, M. W.. Groups generated by reflections and aspherical manifolds not covered by Euclidean space. Ann. of Math. (2) 117 (1983), 293–324.CrossRefGoogle Scholar
[3]Gutkin, E.. Billiard dynamics: a survey with the emphasis on open problems. Regul. Chaotic Dyn. 8 (2003), 1–13.CrossRefGoogle Scholar
[4]Gutkin, E. and Haydn, N.. Topological entropy of polygon exchange transformations and polygonal billiards. Ergod. Th. & Dynam. Sys. 17 (1997), 849–867.CrossRefGoogle Scholar
[5]Gutkin, E. and Tabachnikov, S.. Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards on surfaces of constant curvature. Moscow Math. J. 6 (2006), 673–701.CrossRefGoogle Scholar
[6]Gutkin, E. and Troubetzkoy, S.. Directional flows and strong recurrence for polygonal billiards. Pitman Res. Not. Math. 362 (1996), 21–45.Google Scholar
[7]Hubert, P.. Complexité de suites définies par des billards rationnels. Bull. Soc. Math. Fr. 123 (1995), 257–270.CrossRefGoogle Scholar
[8]Katok, A.. The growth rate for the number of singular and periodic orbits for a polygonal billiard. Comm. Math. Phys. 111 (1987), 151–160.CrossRefGoogle Scholar
[9]Katok, A. and Hasselblatt, B.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.CrossRefGoogle Scholar
[10]Masur, H.. The growth rate of trajectories of a quadratic differential. Ergod. Th. & Dynam. Syst. 10 (1990), 151–176.CrossRefGoogle Scholar