00-519 Lazutkin V.F

A remark on ``Some remarks on the problem of ergodicity of the Standard Map'' (13K, LATeX 2e) Dec 30, 00

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Abstract. We consider the standard family $F_g:(x,y)\mapsto (g\cos(2\pi x)-y,-x)$ of area-preserving maps defined on the two-torus ${\bf R}^2/{\bf Z}^2\,,$ the parameter $g$ ranging along the real axis ${\bf R}\,.$ We prove that there exists a subset ${\cal E}\subset{\bf R}$ whose density tends to zero along the real axis, such that for any $g$ in the complement to ${\cal E}$ the map $F_g$ is ergodic an has nonzero Lyapunov exponents almost everywhere in ${\bf R}^2/{\bf Z}^2$ with respect to the Haar measure.

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