User s g - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:42:05Z http://mathoverflow.net/feeds/user/19462 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/4411/quantitative-versions-of-ergodic-theorem/81669#81669 Answer by S G for Quantitative versions of ergodic theorem S G 2011-11-23T00:00:02Z 2011-11-23T00:00:02Z <p>Usually such estimations require hyperbolicity or some particular kind of system (rotations, IET and similar....).</p> <p>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 <a href="http://arxiv.org/abs/1101.0833" rel="nofollow">1</a> for a very shorth proof of a similar result).</p> <p>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.</p>