Statistical properties of standard maps with increasing coefficient
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.