99-293 Oliver Knill
Positive Kolmogorov-Sinai entropy for the Standard map (295K, LATeX 2e) Aug 3, 99
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Abstract. We prove that the Kolmogorov-Sinai entropy of the Chirikov-Standard map T(x,y)= (2 x-y+ c sin(x),x) with respect to the invariant Lebesgue measure on the two-dimensional torus is bounded below by log(c/2) - C(c) with C(c)=arcsinh(1/c)+log(4/3)/2. For c > c0=3.1547... the entropy of T is positive. This result is stable in Banach spaces of realanalytic symplectic maps: each Chirikov-Standard map with c > c0 is contained in an open set of real-analytic, in general nonergodic area-preserving diffeomorphisms with positive entropy. The Lyapunov exponent estimates hold for a fixed cocycle uniformly for the entire group of measure preserving maps on the torus. This establishes new families of discrete ergodic one-dimensional Schrodinger operators (Lu)(n) = u(n+1) + u(n-1) + c cos(x(n)) u(n) with no absolutely continuous spectrum.

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