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Approximation of commuting transformations


Authors: M. A. Akcoglu and R. V. Chacon
Journal: Proc. Amer. Math. Soc. 32 (1972), 111-119
MSC: Primary 28.70
DOI: https://doi.org/10.1090/S0002-9939-1972-0289745-7
MathSciNet review: 0289745
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Abstract: Let $ \sigma $ and $ \tau $ be two (measure-preserving) transformations. The main purpose of the paper is to show that if $ \tau $ admits approximation by partitions and that if $ \sigma $ commutes with a power $ {\tau ^s}$ of $ \tau $, then $ \sigma $ can be approximated by a finite number of powers of $ \tau $. As an application of the result we solve a problem posed earlier, showing that there exist strongly mixing transformations with only a finite number of prescribed roots.


References [Enhancements On Off] (What's this?)

  • [1] M. A. Akcoglu, R. V. Chacon and T. Schwartzbauer, Commuting transformations and mixing, Proc. Amer. Math. Soc. 24 (1970), 637-642. MR 40 #7421. MR 0254212 (40:7421)
  • [2] R. V. Chacon and T. Schwartzbauer, Commuting point transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 11 (1969), 277-287. MR 39 #2939. MR 0241600 (39:2939)
  • [3] R. V. Chacon, A geometric construction of measure preserving transformations, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), vol. II: Contributions to Probability Theory, part 2, Univ. of California Press, Berkeley, Calif., 1967, pp. 335-360. MR 35 #3033. MR 0212158 (35:3033)
  • [4] N. Friedman, Introduction to ergodic theory, Van Nostrand Math. Studies, New York, 1970. MR 0435350 (55:8310)
  • [5] D. S. Ornstein, A mixing transformation that commutes only with its powers, Proc. Sixth Berkeley Sympos. Math. Statist, and Probability, vol. II, part 2, Univ. of California Press, Berkeley, Calif., 1970, pp. 335-360.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0289745-7
Keywords: Measure-preserving transformations, ergodic theory, commuting transformations
Article copyright: © Copyright 1972 American Mathematical Society

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