A class of ergodic transformations having simple spectrum

Author:
J. R. Baxter

Journal:
Proc. Amer. Math. Soc. **27** (1971), 275-279

MSC:
Primary 28.70; Secondary 47.00

DOI:
https://doi.org/10.1090/S0002-9939-1971-0276440-2

MathSciNet review:
0276440

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Abstract: A class of ergodic, measure-preserving, invertible point transformations is defined, called class . Any measure-preserving point transformation induces a unitary operator on the Hilbert space of -functions. A theorem is proved here which implies that the operator induced by any transformation in class has simple spectrum. [It is then a known result that the transformations in class have zero entropy.]

**[1]**M. A. Akcoglu, R. V. Chacon, and T. Schwartzbauer,*Commuting transformations and mixing*, Proc. Amer. Math. Soc.**24**(1970), 637–642. MR**0254212**, https://doi.org/10.1090/S0002-9939-1970-0254212-1**[2]**R. V. Chacon,*A geometric construction of measure preserving transformations*, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 335–360. MR**0212158****[3]**R. V. Chacon and T. Schwartzbauer,*Commuting point transformations*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**11**(1969), 277–287. MR**0241600**, https://doi.org/10.1007/BF00531651**[4]**Nathaniel A. Friedman,*Introduction to ergodic theory*, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1970. Van Nostrand Reinhold Mathematical Studies, No. 29. MR**0435350****[5]**Donald Ornstein,*A mixing transformation that commutes only with its powers*. (to appear).**[6]**V. A. Rohlin,*Lectures on the entropy theory of transformations with invariant measure*, Uspehi Mat. Nauk**22**(1967), no. 5 (137), 3–56 (Russian). MR**0217258**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1971-0276440-2

Keywords:
Ergodic point transformation,
simple spectrum,
zero entropy,
stacking method

Article copyright:
© Copyright 1971
American Mathematical Society