Let Γ be a subsemigroup of G = GL(d,ℝ), d > 1. We assume that the action of Γ on ${ℝ}^d$ is strongly irreducible and that Γ contains a proximal and quasi-expanding element. We describe contraction properties of the dynamics of Γ on ${ℝ}^d$ at infinity. This amounts to the consideration of the action of Γ on some compact homogeneous spaces of G, which are extensions of the projective space ${\mathbb{P}}^{d-1}$. In the case where Γ is a subsemigroup of GL(d,ℝ) ∩ M(d,ℤ) and Γ has the above properties, we deduce that the Γ-orbits...

We study shadowing properties of continuous actions of the groups ${\mathbb{Z}}^p$ and ${\mathbb{Z}}^p\times {ℝ}^p$. Necessary and sufficient conditions are given under which a linear action of ${\mathbb{Z}}^p$ on ${ℂ}^m$ has a Lipschitz shadowing property.

Building on earlier work of Katětov, Uspenskij proved in [8] that the group of isometries of Urysohn's universal metric space 𝕌, endowed with the pointwise convergence topology, is a universal Polish group (i.e. it contains an isomorphic copy of any Polish group). Answering a question of Gao and Kechris, we prove here the following, more precise result: for any Polish group G, there exists a closed subset F of 𝕌 such that G is topologically isomorphic to the group of isometries of 𝕌 which map...