Dynamics of non abelian affine homotheties group of C^n

TL;DR

This paper investigates the dynamics of non-abelian groups generated by affine homotheties acting on complex n-space, revealing the structure of orbit closures and conditions for density and minimality.

Contribution

It characterizes the orbit closures of non-abelian affine homothety groups on C^n, identifying invariant subspaces and subgroup structures that determine orbit behavior.

Findings

Orbit closures are affine transformations of a subgroup H and an invariant subspace E.

Orbits in E are dense, indicating complex dynamical behavior.

Orbits outside E are minimal, showing different dynamical regimes.

Abstract

In this paper we study the action of non abelian subgroup G generated by affine homotheties on C^n. We prove that there exist a subgroup H of C\{0}, a G-invariant affine subspace E of C^n and b in E such that the closure of any orbit G(z) is equal to H(z-a)+E, z in C^n. In particular, every orbit in E is dense in it. Moreover, if the complementary U=C^n \E is non empty, every orbit of U is minimal in it.