Absence of mixing for smooth flows on genus two surfaces
Dmitri Scheglov
Journal of Modern Dynamics 2009, 3(1): 13-34 doi: 10.3934/jmd.2009.3.13
We prove that typical area-preserving flows with linearly isomorphic nondegenerate saddles on genus two surfaces are not mixing.
keywords: mixing. Rauzy induction interval-exchange transformations
Growth of periodic orbits and generalized diagonals for typical triangular billiards
Dmitri Scheglov
Journal of Modern Dynamics 2013, 7(1): 31-44 doi: 10.3934/jmd.2013.7.31
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.

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