Statistical properties of standard maps with increasing coefficient
Alex Blumenthal

The standard map (with large enough coefficient) is an area preserving 
map of the torus exhibiting hyperbolicity on a large but noninvariant 
subset of phase space: as the coefficient increases, the size of this 
"good region" increases, as does the strength of hyperbolicity on this
subset. In spite of this, it is notoriously difficult to determine 
asymptotic properties of iterates of the standard map (e.g. metric 
entropy), even for large coefficient values for which the dynamics 
should be highly chaotic.

If, on the other hand, we allow the coefficient of the standard map to 
increase with time, the resulting (nonautonomous) composition of maps is 
increasingly "predominantly" hyperbolic with time. If the growth rate of 
the coefficients is sufficiently rapid, e.g. polynomial of order > 1, 
this nonautonomous composition exhibits sensitivity to initial 
conditions (in the sense of positive Lyapunov exponents) by a simple 
Borel-Cantelli argument. A natural question, then, is to go farther and 
obtain quantitative statistical properties. In this setting, I will 
present decay of correlations estimates and (for sufficiently rapidly 
increasing coefficients) a CLT for Holder observables.

This work is joint with Dmitry Dolgopyat.