Berend, Daniel Multi-invariant sets on tori. (English) Zbl 0532.10028 Trans. Am. Math. Soc. 280, 509-532 (1983). 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 Cited in 7 ReviewsCited in 26 Documents MSC: 11K60 Diophantine approximation in probabilistic number theory 28D15 General groups of measure-preserving transformations 54H15 Transformation groups and semigroups (topological aspects) Keywords:semigroups of endomorphisms of tori; invariant sets; ID-property; diophantine approximations Citations:Zbl 0146.285 PDFBibTeX XMLCite \textit{D. Berend}, Trans. Am. Math. Soc. 280, 509--532 (1983; Zbl 0532.10028) Full Text: DOI 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 [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. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.