English

Pointwise convergence along cubes for measure preserving systems

Dynamical Systems 2007-05-23 v1

Abstract

Let (X,B,μ)(X, \mathcal{B}, \mu) be a probability measure space and T1T_1, T2T_2, T3T_3 three not necessarily commuting measure preserving transformations on (X,B,μ)(X, \mathcal{B}, \mu). We prove that for all bounded functions f1f_1, f2f_2, f3f_3 the averages 1N2n,m=1Nf1(T1nx)f2(T2mx)f3(T3n+mx)\frac{1}{N^2}\sum_{n, m =1}^N f_1(T_1^nx)f_2(T_2^mx)f_3(T_3^{n+m}x) converges a.e. Generalizations to averages of 2k12^k -1 functions are also given for not necessarily commuting weakly mixing systems.

Keywords

Cite

@article{arxiv.math/0311274,
  title  = {Pointwise convergence along cubes for measure preserving systems},
  author = {Idris Assani},
  journal= {arXiv preprint arXiv:math/0311274},
  year   = {2007}
}

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14 pages