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Multi-invariant sets on tori. (English) Zbl 0532.10028
Let $$\Sigma$$ be a commutative semigroup of continuous endomorphisms of the r-dimensional torus $$T^ r$$, $$r\geq 2$$. $$\Sigma$$ is said to have the ID-property if the only infinite, closed, $$\Sigma$$-invariant subset of $$T^ r$$ is $$T^ r$$ itself. The author proves that $$\Sigma$$ has the ID- property iff the following conditions are satisfied: (1) There exists $$\sigma\in \Sigma$$ such that the characteristic polynomial of $$\sigma^ n$$ is irreducible over $${\mathbb{Z}}$$ for every positive integer n. (2) For every common eigenvector $$v\in {\mathbb{C}}^ r$$ of $$\Sigma$$ there exists $$\sigma\in \Sigma$$ such that the corresponding eigenvalue $$\lambda\in {\mathbb{C}}$$ satisfies $$| \lambda |>1$$. (3) $$\Sigma$$ contains a pair of rationally independent endomorphisms (i.e. there are $$\sigma$$,$$\tau\in \Sigma$$ such that $$\sigma^{\ell}\neq \tau^ m$$ for all $$(\ell,m)\in {\mathbb{Z}}^ 2\backslash(0,0)).$$
This generalizes a result by H. Furstenberg (for $$r=1)$$ [Math. Syst. Theory 1, 1-49 (1967; Zbl 0146.285)]. The proof uses ideas from algebraic number theory and diophantine approximations and gives also some further informations on the semigroups described above.
Reviewer: V.Losert

##### MSC:
 11K60 Diophantine approximation in probabilistic number theory 28D15 General groups of measure-preserving transformations 54H15 Transformation groups and semigroups (topological aspects)
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##### References:
 [1] Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1 – 49. · Zbl 0146.28502 · doi:10.1007/BF01692494 · doi.org [2] A. Y. Khinchin, Three pearls of number theory, Graylock Press, Rochester, N. Y., 1952. · Zbl 0048.27202 [3] Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, PWN — Polish Scientific Publishers, Warsaw, 1974. Monografie Matematyczne, Tom 57. · Zbl 0276.12002 [4] Kenneth B. Stolarsky, Algebraic numbers and Diophantine approximation, Marcel Dekker, Inc., New York, 1974. Pure and Applied Mathematics, No. 26. · Zbl 0285.10022 [5] B. L. van der Waerden, Modern algebra, vol. I, Ungar, New York, 1953.
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