Polygonal billiards have zero topological entropy. Complexity is a finer measure of their asymptotic behavior. In this article we show an explicit quadratic lower bound for the complexity of the billiard in an arbitrary polygon.
REFERENCES
1.
G.
Galperin
, T.
Krüger
, and S.
Troubetzkoy
, “Local instability of orbits in polygonal and polyhedral billiards
,” Commun. Math. Phys.
169
, 463
–473
(1995
).2.
A.
Katok
, “The growth rate for the number of singular and periodic orbits for a polygonal billiard
,” Commun. Math. Phys.
111
, 151
–160
(1987
).3.
4.
E.
Gutkin
, “Billiard in polygons: Survey of recent results
,” J. Stat. Phys.
174
, 43
–56
(1995
).5.
S. Tabachnikov, Billiards, “Panorames et syntheses,” Soc. Math. France (1995).
6.
H. Masur, “The growth rate of trajectories of a quadratic differential,” Erg. Th. Dyn. Sys. 10, 151–176 (1990).
7.
Ya. Vorobets, “Ergodicity of billiards in polygons,” Sbornik: Math., 389–434 (1997).
8.
P. Hubert, “Compexité de suites définies par des trajectoires de billiard,” Bull. Soc. Math. France 123, 257–270 (1995).
9.
P. Hubert, “Dynamique symbolique des billards polygonaux rationnels,” Thesis, Universite de Marseilles, 1995.
10.
E. Gutkin and S. Troubetzkoy, “Directional flows and strong recurrence for polygonal billiards,” in Proceedings of the International Congress of Dynamical Systems, Montevideo, Uruguay, F. Ledrappier, edited by J. Lewowicz and S. Newhouse (Addison-Wesley, New York, 1997).
11.
P. Arnoux, C. Mauduit, I. Shiokawa, and J. Tamura, “Complexity of sequences defined by billiards in a cube,” Bull. Soc. Math. France 119 1–12 (1991).
12.
Y.
Baryshnikov
, “Complexity of trajectories in rectangular billiards
,” Commun. Math. Phys.
174
, 43
–56
(1995
).
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© 1998 American Institute of Physics.
1998
American Institute of Physics
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