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Semigroup actions on tori and stationary measures on projective spaces. (English) Zbl 1087.37022

A multiplicative semigroup \(\Gamma\) of integers is said to be lacunary if the members \(\{n \in \Gamma : \, n > 0\}\) are of the form \(n_0^k\), \(k \in \mathbb N\), \(n_0 \in \mathbb N^*\). In 1914, Hardy and Littlewood proved that if \(r\) is a positive integer and \(\alpha\) is an irrational number, then the set \(\{q^r \alpha: \, q \in \mathbb N\}\) is dense modulo 1. In 1967, Furstenberg generalized this result, showing that if \(\Gamma\) is a non-lacunary semigroup of integers and \(\alpha\) is an irrational number, then the orbit \(\Gamma \alpha\) is dense modulo 1.
In this paper, the authors extend these statements to several variables, by replacing the semigroup \(\mathbb Z^*\) by \(M_{\text{inv}} (d,\mathbb Z) := \text{GL}(d,\mathbb Z) \cap M(d,\mathbb Z)\), \(d > 1\), where \(\text{GL}(d,\mathbb R)\) is the group of invertible \(d \times d\)-matrices with real entries and \(M(d,\mathbb Z)\) is the set of \(d \times d\)-matrices with integer entries, acting by endomorphisms on the torus \(\mathbb T^d :=\mathbb R^d/\mathbb Z^d\). A matrix \(A \in \text{GL}(d,\mathbb R)\) is said to be proximal if it has an eigenvalue \(\lambda_A\) such that \(| \lambda_A| > | \lambda | \) for all other eigenvalues \(\lambda\) of \(A\). A matrix \(A\) is said to be quasi-expanding if it has an eigenvalue \(\lambda\) such that \(| \lambda | > 1\). A subsemigroup \(\Gamma\) of \(\text{GL}(d,\mathbb R)\) is said to be strongly irreducible if no finite union of proper subspaces of \(\mathbb R\) is \(\Gamma\)-invariant.
The main result in this paper is the following: Let \(\Gamma\) be a strongly irreducible subsemigroup of \(M_{\text{inv}} (d,\mathbb Z)\), \(d > 1\), such that \(\Gamma\) contains a proximal and quasi-expanding element. Then the semigroup \(\Gamma\) acting on \(\mathbb T^d\) has the property that every infinite \(\Gamma\)-invariant subset of \(\mathbb T^d\) is dense. (The authors also prove that under strong reducibility and proximality, the existence of a proximal and quasi-expanding element in \(\Gamma\) is equivalent to the unboundedness of \(\Gamma\)-orbits in \(\mathbb R \setminus \{0\}\).) The underlying idea for the proof is to lift the automorphisms of \(\mathbb T^d\) to its universal cover \(\mathbb R^d\) and to study the action of the lifts at infinity, so describing the contraction properties of the dynamics of \(\Gamma\) at infinity. This action can be expressed in terms of some compact homogeneous spaces of \(\text{GL}(d,\mathbb R)\) which are closely related to the projective space \(\mathbb P^{d-1}\). Random walks techniques are used to consider the global semigroup asymptotic behavior in terms of stationary measures and convergence to them.
A corrigendum has been published in Stud. Math. 183, No. 2, 195–196 (2007; Zbl 1196.37009).

MSC:

37A15 General groups of measure-preserving transformations and dynamical systems
22F05 General theory of group and pseudogroup actions
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
54H20 Topological dynamics (MSC2010)
60B11 Probability theory on linear topological spaces

Citations:

Zbl 1196.37009
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