Mathematics – Dynamical Systems
Scientific paper
2009-10-06
Mathematics
Dynamical Systems
Part two of three. 103 pages. [TDA Nov 11th 2009:] Several mistakes corrected. The Main Example of Section 4 in v1 has been re
Scientific paper
In this paper we combine the general tools developed in (arXiv:0905.0518) with several ideas taken from earlier work on one-dimensional nonconventional ergodic averages by Furstenberg and Weiss, Host and Kra and Ziegler to study the averages \frac{1}{N}\sum_{n=1}^N(f_1\circ T^{n\bf{p}_1})(f_2\circ T^{n\bf{p}_2})(f_3\circ T^{n\bf{p}_3}) for f_1,f_2,f_3 \in L^\infty(\mu) associated to a triple of directions \bf{p}_1,\bf{p}_2,\bf{p}_3 \in \bbZ^2 that lie in general position along with 0 \in \bbZ^2. We will show how to construct a `pleasant' extension of an initially-given \bbZ^2-system for which these averages admit characteristic factors with a very concrete description, involving one-dimensional isotropy factors and two-step pro-nilsystems. We also use this analysis to construct pleasant extensions and then prove norm convergence for the polynomial nonconventional ergodic averages \frac{1}{N}\sum_{n=1}^N(f_1\circ T_1^{n^2})(f_2\circ T_1^{n^2}T_2^n) associated to two commuting transformations T_1, T_2.
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