Journal of Modern Dynamics (JMD)

Growth of periodic orbits and generalized diagonals for typical triangular billiards

Pages: 31 - 44, Issue 1, March 2013      doi:10.3934/jmd.2013.7.31

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Dmitri Scheglov - Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, United States (email)

Abstract: 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: 37C35.

Received: May 2012;      Revised: October 2012;      Available Online: May 2013.