 99293 Oliver Knill
 Positive KolmogorovSinai entropy for the Standard map
(295K, LATeX 2e)
Aug 3, 99

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Abstract. We prove that the KolmogorovSinai entropy of the
ChirikovStandard map T(x,y)= (2 xy+ c sin(x),x) with respect to the
invariant Lebesgue measure on the twodimensional 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 ChirikovStandard map with c > c0 is contained in an
open set of realanalytic, in general nonergodic areapreserving
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
onedimensional Schrodinger operators
(Lu)(n) = u(n+1) + u(n1) + c cos(x(n)) u(n)
with no absolutely continuous spectrum.
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