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Growth of periodic orbits and generalized diagonals for typical triangular billiards

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  • 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.
    Mathematics Subject Classification: Primary: 37C10; Secondary: 37C35.


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