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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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[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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