We give a characterization of hypercyclic abelian group $\mathcal{G}$ of affine maps on $\mathbb{C}^{n}$. If $\mathcal{G}$ is finitely generated, this characterization is explicit. We prove in particular that no abelian group generated by $n$ affine maps on $\mathbb{C}^{n}$ has a dense orbit.

Keywords:

affine, hypercyclic, dense, orbit, affine group, abelian affine, hypercyclic, dense, orbit, affine group, abelian

MSC Classifications:

37C85, 47A16 show english descriptions Dynamics of group actions other than ${\bf Z}$ and ${\bf R}$, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx]
Cyclic vectors, hypercyclic and chaotic operators 37C85 - Dynamics of group actions other than ${\bf Z}$ and ${\bf R}$, and foliations [See mainly 22Fxx, and also 57R30, 57Sxx]
47A16 - Cyclic vectors, hypercyclic and chaotic operators

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