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 and be two (measure-preserving) transformations. The main purpose of the paper is to show that if admits approximation by partitions and that if commutes with a power of , then can be approximated by a finite number of powers of . 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.

**[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