On the action of the semigroup of non singular integral matrices on $\R^n$

Abstract.

Let Γ be the multiplicative semigroup of all n × n matrices with integral  entries and nonzero determinant. Let 1 ≤ p ≤ n−1 and V = Rnp = Rn ⊕···⊕Rn (p copies). Consider the action of Γ on V , given by the natural action on each component, by matrix multiplication on the left. Then for x= (x1, . . . , xp) ∈ V , the Γ-orbit is dense

in V if and only if there is no linear combination pj=1 λjxj, with λj ̸= 0 for some j, which is a rational vector in Rn; in fact the assertion holds also for the orbit of the subgroup SL(n, Z) that is contained in Γ. When x is such that the Γ-orbit is dense, given y ∈ Rnp, and ǫ > 0 one may ask for γ ∈ Γ such that |γx−y| < ǫ, with a bound on |γ| in terms of ǫ.

There has been considerable interest in the literature in quantitative results of this kind, for various group actions. In particular it was shown by Laurent-Nogueira, for n = 2, that given an irrational vector x in R2, any target vector y ∈ R2 and ρ < 1 there exist 3 infinitely many γ in SL(2, Z) such that |γx − y| ≤ |γ|−ρ. In the talk we will describe some results along this theme for the action of Γ; for the case n = 2 the result is stronger in import than what is recalled above for SL(2, Z), in the sense that the corresponding statement holds for all ρ less than 1, in place of 1 for SL(2, Z).

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The talk is based on a joint work with S.G. Dani.

Posted on Nov 19, 2018 in Dynamical Systems, Seminars