Powers of sequences and convergence of ergodic averages

Abstract : A sequence $(s_n)$ of integers is good for the mean ergodic theorem if for each invertible measure preserving system $(X,\mathcal{B},\mu,T)$ and any bounded measurable function $f$, the averages $ \frac1N \sum_{n=1}^N f(T^{s_n}x)$ converge in the $L^2$ norm. We construct a sequence $(s_n)$ that is good for the mean ergodic theorem, but the sequence $(s_n^2)$ is not. Furthermore, we show that for any set of bad exponents $B$, there is a sequence $(s_n)$ where $(s_n^k)$ is good for the mean ergodic theorem exactly when $k$ is not in $B$. We then extend this result to multiple ergodic averages. We also prove a similar result for pointwise convergence of single ergodic averages.
Document type :
Journal articles
Complete list of metadatas

Cited literature [14 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-00327604
Contributor : Emmanuel Lesigne <>
Submitted on : Monday, June 29, 2009 - 4:36:43 PM
Last modification on : Thursday, August 22, 2019 - 1:13:17 AM
Long-term archiving on : Friday, September 24, 2010 - 11:20:00 AM

Files

good-and-bad-conv12.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-00327604, version 4
  • ARXIV : 0810.1581

Collections

Citation

Nikos Frantzikinakis, Michael Johnson, Emmanuel Lesigne, Mate Wierdl. Powers of sequences and convergence of ergodic averages. Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2010, 30 (5), pp.1431-1456. ⟨hal-00327604v4⟩

Share

Metrics

Record views

220

Files downloads

175