# American Institute of Mathematical Sciences

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

## Growth of periodic orbits and generalized diagonals for typical triangular billiards

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

Received  May 2012 Revised  October 2012 Published  May 2013

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.
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
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