Action of Non Abelian Group Generated by Affine Homotheties on R^n

TL;DR

This paper investigates how non-abelian groups generated by affine homotheties act on R^n, revealing two main types of orbit closures related to invariant subspaces and subgroup unions.

Contribution

It characterizes the orbit closure structures of non-abelian affine homothety groups on R^n, providing a detailed classification of their dynamical behavior.

Findings

Orbit closures are either affine subspace-based or unions of subgroup translates.

G(x) is dense in certain invariant subspaces for all x in those subspaces.

Every orbit outside a specific invariant subspace is minimal in its complement.

Abstract

In this paper, we study the action of non abelian group G generated by affine homotheties on R^n. We prove that G satisfies one of the following properties: (i) there exist a subgroup F_{G} of R\{0} containing 0 in its closure, a G-invariant affine subspace E_{G} of R^n and a in E_{G} such that for every x in R^n the closure of the orbit G(x) is equal to F_{G} .(x - a) +E_{G}. In particular, G(x) is dense in E_{G} for every x in E_{G} and every orbit of U = R^n\E_{G} is minimal in U. (ii) there exists a closed subgroup H_{G} of R^n and a in R^n such that for every x in R^n, the closure of the orbit G(x) is equal to the union of (x + H_{G}) and (-x + a + H_{G}).