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?