Journal of Modern Dynamics (JMD)

Lower bounds on growth rates of periodic billiard trajectories in some irrational polygons

Pages: 649 - 663, Issue 4, October 2007      doi:10.3934/jmd.2007.1.649

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W. Patrick Hooper - Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL, 60208-2730, United States (email)

Abstract: In this paper we show that there exist irrational polygons $P$ where the number of periodic billiard paths of length less than $n$, $f_P(n)$, grows superlinearly. In fact, if we fix the number of sides of our polygon, for any $k \in \N$ there is an open set of polygons where $f_P(n)$ grows faster than $n \log^k n$.

Keywords:  irrational billiards, periodic billiard paths.
Mathematics Subject Classification:  Primary: 37D50; Secondary: 37E15, 37E35.

Received: April 2007;      Revised: July 2007;      Available Online: July 2007.