Let $U$ be a contractible bounded open subset of $\Bbb{C}$. There is a standard classification of possible dynamical behaviors of holomorphic maps $f:U\rightarrow U$:
- Attracting Case: There is an attracting fixed point in $U$ to which all orbits converge.
- Escape: For any compact subset $K$ of $U$, the intersection $f^{\circ n}(K)\cap K$ is empty for $n$ large enough.
- Finite Order: Some iterate of $f$ is identity.
- Irrational Rotation: There is an $\alpha\in\Bbb{R}-\Bbb{Q}$ such that $f$ is conjugate to $\Bbb{D}\rightarrow\Bbb{D}:z\mapsto{\rm{e}}^{2\pi{\rm{i}}\alpha}z$.
A proof can be found for example in Milnor's book on complex dynamics.
My question: What is known or can be said when $U$ is a contractible bounded open subset of $\Bbb{C}^n$? Are there examples of holomorphic maps $f:U\rightarrow U$ that exhibit a different dynamical behaviors; that is, $f$ is of infinite order, is not conjugate to a linear map, does not have any attracting periodic point, and admits orbits with accumulation points inside $U$.
