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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Multi-invariant sets on tori
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by Daniel Berend PDF
Trans. Amer. Math. Soc. 280 (1983), 509-532 Request permission


Given a compact metric group $G$, we are interested in those semigroups $\Sigma$ of continuous endomorphisms of $G$, possessing the following property: The only infinite, closed, $\Sigma$-invariant subset of $G$ is $G$ itself. Generalizing a one-dimensional result of Furstenberg, we give here a full characterization—for the case of finitedimensional tori—of those commutative semigroups with the aforementioned property.
  • Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 213508, DOI 10.1007/BF01692494
  • A. Y. Khinchin, Three pearls of number theory, Graylock Press, Rochester, N.Y., 1952. MR 0046372
  • Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, Monografie Matematyczne, Tom 57, PWN—Polish Scientific Publishers, Warsaw, 1974. MR 0347767
  • Kenneth B. Stolarsky, Algebraic numbers and Diophantine approximation, Pure and Applied Mathematics, No. 26, Marcel Dekker, Inc., New York, 1974. MR 0374041
  • B. L. van der Waerden, Modern algebra, vol. I, Ungar, New York, 1953.
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Additional Information
  • © Copyright 1983 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 280 (1983), 509-532
  • MSC: Primary 11K06; Secondary 11K55, 28D10, 54A15
  • DOI:
  • MathSciNet review: 716835