Hypercyclicity and k-Transitivity (k>=2) for abelian semigroup of affine maps on C^n

TL;DR

This paper establishes the minimal number of affine maps needed for hypercyclicity in abelian semigroups on complex n-space and proves such groups cannot be k-transitive for k>=2.

Contribution

It proves that n+1 affine maps are necessary for hypercyclicity and that finitely generated abelian groups of affine maps are never k-transitive for k>=2.

Findings

Minimal number of affine maps for hypercyclicity is n+1.

Finitely generated abelian groups of affine maps are not k-transitive for k>=2.

Provides conditions for hypercyclicity in affine semigroups.

Abstract

In this paper, we prove that the minimal number of affine maps on C^n, required to form a hypercyclic abelian semigroup on C^n is n+1. We also prove that the action of any abelian group finitely generated by affine maps on C^n, is never k-transitive for k>=2.