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Semigroup action on \(\mathbb T^n\). (English) Zbl 1071.37008

Author’s abstract: Let \(S\) be a semigroup of nonsingular \(n\times n\)-matrices with integer coefficients. There is a natural action of \(S\) on the \(n\)-dimensional torus \(\mathbb{T}^n\). We give a complete characterization of \(S\) that satisfies the following property (ID): The only infinite closed \(S\)-invariant subset of \(\mathbb{T}^n\) is \(\mathbb{T}^n\) itself. We prove that the semigroup of affine transformations, whose linear parts satisfy property ID, also satisfies property ID. This generalizes the results of H. Fürstenberg for a circle and D. Berend for commutative semigroups. In addition, we describe orbits for semigroups that are not virtually cyclic and act strongly irreducibly on \(\mathbb{T}^n\). We also give a description of orbits under actions of nonvirtually cyclic irreducible semigroups. Furthermore, we obtain a characterization of closed minimal sets of such actions and we prove that an irreducible subgroup of \(\text{SL}(n,\mathbb{Z})\) acts tautly on \(\mathbb{T}^n\).

MSC:

37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
47D03 Groups and semigroups of linear operators
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
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