Oliver Knill
Positive Kolmogorov-Sinai entropy for the Standard map
(295K, LATeX 2e)

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.