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

TL;DR

This paper investigates the dynamics of non-abelian groups generated by affine homotheties on R^n, revealing their orbit closures and invariant structures, which depend on specific subgroup configurations.

Contribution

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

Findings

Orbit closures are either affine subspace-based or unions of translated subgroups.

Existence of invariant affine subspaces or subgroup unions depending on the group structure.

Minimality of orbits in certain invariant subsets.

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}).