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On convergence of the averages\(\frac{1}{N}\sum\nolimits_{n = 1}^N {f_1 (R^n x)f_2 (S^n x)f_3 (T^n x)} \)

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Abstract

In this note, we will prove that for commuting ergodic measure preserving transformationsR, S andT, ifRT −1,ST −1 are also ergodic, then the limit

$$\lim \frac{1}{N}\sum\nolimits_{n = 1}^N {f_1 (R^n x)f_2 (S^n x)f_3 (T^n x)} $$

exists inL 1-norm. The method used in this note was developed byConze, Furstenberg, Lesigne andWeiss.

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References

  1. Bergelson, V.: Weakly mixing pet. Ergodic Th. Dynamical System7, 337–349 (1987).

    Google Scholar 

  2. Bourgain, J.: Double recurrence and almost sure convergence. J. Reine Angew. Math.404, 140–161 (1987).

    Google Scholar 

  3. Conze, J., Lesigne, E.: Théorèmes ergodiques pour des mesures diagonales. Bull. Soc. Math. France112, 143–175 (1984).

    Google Scholar 

  4. Conze, J., Lesigne, E.: Sur un théorème ergodique pour des mesures diagonales. Publications de l'IRMAR. Probabilities. Univ. Rennes (France)87. Abstract in: C. R. Acad. Sci. Paris306, Serie I 491–493 (1988).

  5. Furstenberg, H.: Ergodic behavior of diagonal measures and a theorem of Szemeredi on arithmetic progressions. J. d'Analyse Math31, 204–256 (1977).

    Google Scholar 

  6. Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton, N.J.: University Press. 1981.

    Google Scholar 

  7. Furstenberg, H.: Nonconventional ergodic averages. The Legacy of John von Neumann (Hempstead, NY, 1988), 43–56. Proc. Sympos. Pure Math., 50. Providence, RI: Amer. Math. Soc. 1990.

    Google Scholar 

  8. Furstenberg, H., Weiss, B.: Private communication.

  9. Lesigne, E.: Resolution d'une équation fonctionelle. Bull. Soc. Math. France112, 176–196 (1984).

    Google Scholar 

  10. Lesigne, E.: Théorèmes ergodiques pour une translation sur un nilvariété. Ergodic Th. and Dynamical System9, 115–126 (1989).

    Google Scholar 

  11. Raghunathan, M. S.: Discrete subgroups of Lie groups. New York: Springer. 1972.

    Google Scholar 

  12. Rudolph, D. J.: Eigenfunctions ofT×S and Conze-Lesigne Algebra. To appear in Proceeding of Alexandia Conference on Ergodic Theory and Its Connections with Harmonic Analysis (1993). Cambridge: University Press.

  13. Zimmer, R.: Extensions of ergodic group actions. Illinois J. Math.20, 373–409 (1976).

    Google Scholar 

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  1. Department of Mathematics, Clark College, 1800 McLoughlin Blvd, 98663, WA, Vancouver, USA

    Qing Zhang

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  1. Qing Zhang
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Zhang, Q. On convergence of the averages\(\frac{1}{N}\sum\nolimits_{n = 1}^N {f_1 (R^n x)f_2 (S^n x)f_3 (T^n x)} \) . Monatshefte für Mathematik 122, 275–300 (1996). https://doi.org/10.1007/BF01320190

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  • DOI: https://doi.org/10.1007/BF01320190

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