Convergence of diagonal ergodic averages. Towsner, H. *Ergodic Theory Dynam. Systems*, 29(4):1309–1326, 2009. Journal Arxiv doi abstract bibtex 21 downloads Tao has recently proved that if T_{1},…,T_{l} are commuting, invertible, measure-preserving transformations on a dynamical system then for any L_{∞} functions f_{1},…f_{l}, the average (1/N)Σ_{n=0}^{N-1}Π_{i≤l}f_{i} o T^{in} converges in the L_{2} norm. Tao's proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence "backwards". In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao's argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties.

@article {MR2529651,
AUTHOR = {Towsner, Henry},
TITLE = {Convergence of diagonal ergodic averages},
JOURNAL = {Ergodic Theory Dynam. Systems},
FJOURNAL = {Ergodic Theory and Dynamical Systems},
VOLUME = {29},
YEAR = {2009},
NUMBER = {4},
PAGES = {1309--1326},
ISSN = {0143-3857},
MRCLASS = {37A30 (28D15)},
MRNUMBER = {2529651 (2010f:37005)},
MRREVIEWER = {Bryna Kra},
DOI = {10.1017/S0143385708000722},
URLJOURNAL= {http://dx.doi.org/10.1017/S0143385708000722},
urlarxiv={http://arxiv.org/abs/0711.1180},
abstract={Tao has recently proved that if T<sub>1</sub>,…,T<sub>l</sub> are commuting, invertible, measure-preserving transformations on a dynamical system then for any L<sub>∞</sub> functions f<sub>1</sub>,…f<sub>l</sub>, the average (1/N)Σ<sub>n=0</sub><sup>N-1</sup>Π<sub>i≤l</sub>f<sub>i</sub> o T<sup>in</sup> converges in the L<sub>2</sub> norm. Tao's proof is unusual in that it translates the problem into a more complicated statement about the combinatorics of finite spaces by using the Furstenberg correspondence "backwards". In this paper, we give an ergodic proof of this theorem, essentially a translation of Tao's argument to the ergodic setting. In order to do this, we develop two new variations on the usual Furstenberg correspondence, both of which take recurrence-type statements in one dynamical system and give equivalent statements in a different dynamical system with desirable properties.
},
}

Downloads: 21

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