Are there any general theorems similar to Birkhoff's ergodic theorem, but giving quantitative estimates on the rate of convergence or average time of recurrence (perhaps with additional assumptions)? Take the example of an "irrational rotation" on the unit circle – are there any estimates on the average time it takes for a point to hit a certain interval?
I know there are such theorems for very special systems (e.g. for Markov chains we have exponential convergence) – what can be said about a "generic" ergodic system?
Best Answer
The best effective estimate I know of in general is the very recent and impressive work of Einsiedler-Margulis-Venkatesh. They provide polynomial decay in the ergodic theorem (for unipotent and other flows) on a set of large (actually estimated) measure for quotients of most lie groups by arithmetic lattices.
There were previously things like this: for closed hyperbolic surfaces, the error for the time average of the horocycle flow of length $T$ is something like $S(f) T^{-\epsilon}$ where $f$ is the smooth function you're averaging, $S(f)$ is a Sobolev norm, and $\epsilon$ depends on the spectrum of the surface. (In this case, every point is generic for the horocycle flow (by Ratner's theorem), and this effective estimate is independent of the point you use for the average!)
See the nice survey "An introduction to effective equidistribution and property (tau)," by Einsiedler, found on his webpage.
I would be very interested in any better estimates for flows on arithmetic hyperbolic $3$-manifolds given information on the trace field, quaternion algebra, et cetera.