Growth of periodic orbits and generalized diagonals for typical triangular billiards
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.
Received: May 2012; Revised: October 2012; Available Online: May 2013. |