Abstract
Let S be a semigroup of nonsingular n × 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. Furstenberg 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 action 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 SL(n, ℤ) acts tautly on \({\mathbb{T}^n}\).
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Mathematics Subject Classifications (2000). Primary: 37B05; secondary: 47D03, 57S05, 57S25.
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Muchnik, R. Semigroup Actions on \({\mathbb{T}^n}\). Geom Dedicata 110, 1–47 (2005). https://doi.org/10.1007/s10711-004-4321-7
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DOI: https://doi.org/10.1007/s10711-004-4321-7