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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?

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6 Answers 6

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.

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The survey link is missing but thanks for the pointer! – Kaveh Khodjasteh Sep 20 '10 at 14:37
Sorry about the link rot. I edited to correct it. Let's hope it lasts a little longer this time. – Richard Kent Sep 20 '10 at 14:57

For the specific case you mention of an irrational rotation of the circle, it depends on the rotation number $r$. You can get slow asymptotic convergence for something like $$r=\sum{10^{-n!}}$$ (and you can just keep adding factorial signs to make the convergence arbitrarily slow.)

At the other end of the spectrum is rotation number equal to the golden ratio. In this case you know that for a Fibonacci number $q$, it will only take $q$ iterations to hit each interval of the form $[n/q, (n+1)/q]$.

The behavior for an arbitrary irrational number is governed by its continued fraction expansion. Milnor's book Dynamics in one complex variable has a clear explanation.

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The survey "The rate of convergence in ergodic theorems" by A. G. Kachurovskii (Russian Math. Surveys 1996) lists quite a number of results in this direction which might be of interest to you. It includes some negative results: for example, if I recall correctly, for any positive real sequence a_n=o(n) and any aperiodic measurable dynamical system, we can find a measurable function f taking only two distinct real values with the property that the ergodic sums of f are o(n) but not o(a_n). On the other hand, a number of sufficient conditions are presented for polynominal error estimates to hold in the ergodic theorem.

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A. Leibman proved a quantitative lower bound for the averages $\frac{1}{N}\sum_{n=0}^{N-1} \mu(A\cap T^{-n}A)$ in terms of $\mu(A)$ (note: the sum begins at $n=0$). The bound is $$ \frac{1}{N}\sum_{n=0}^{N-1} \mu(A\cap T^{-n}A) \geq \sqrt{\mu(A)^2+(1-\mu(A))^2} + \mu(A)-1 $$ for all $N\geq 1$ when $T$ is a measure preserving transformation of a probability space, and this is the best possible such bound.

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Dolpogyat has proved rates of convergence for Anosov systems (see #4 here.)

Kac pioneered the study of recurrence times for stochastic processes (see here).

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Usually such estimations require hyperbolicity or some particular kind of system (rotations, IET and similar....).

For general systems an effective quantitative estimation on the rate of convergence is possible (altough not sharph), provided that the system is given effectively.(see J. Avigad, P. Gerhardy and H. Towsner, Local stability of ergodic averages, Transactions of the American Mathematical Society, 362 (2010), or 1 for a very shorth proof of a similar result).

In your question it seems to me that you are mostly interested to the behavior of hitting times, and perhaps in rotations. As already remarked in another answer this depend on the arithmetical properties of the rotation. If the rotation has an angle which is well approximated by rationals you can have long hitting times for certain intervals (the time you need to wait for the interval to be hit is much more than the inverse of the lenght of the interval). Quantitative convergence results are given by the so called "discrepancy" estimations, in function of the Diofantine type of the angle (these are classical results you can find in many books). If you consider multidimensional rotations you can have even stronger pathological behaviors of the hitting time, but I do not know if you are interested in this.

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