Powers of sequences and convergence of ergodic averages
نویسنده
چکیده
A sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure preserving system (X, B, µ, T) and any bounded measurable function f , the averages 1 N P N n=1 f (T sn x) converge in the L 2 (µ) norm. We construct a sequence (sn) that is good for the mean ergodic theorem, but the sequence (s 2 n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (s k n) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form 1 N P N n=1 f 1 (T sn x)f 2 (T 2sn x). .. f ℓ (T ℓsn x). We also prove a similar result for pointwise convergence of single ergodic averages.
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