Flow invariant Runge domains and global linearization of holomorphic vector fields

In this paper, we study two problems concerning holomorphic flows on $\mathbb C^n$. First, we prove Runge-type results for positive-time flow invariant domains. For a linear flow $e^{tA}$, where $A\in GL(n,\mathbb C)$, let $E^s$, $E^u$, and $E^c$ denote the stable, unstable, and center subspaces of $A$, respectively. We show that if a positive-time flow invariant domain $\Omega\subset\mathbb C^n$ contains the origin and the center subspace, and if $E^u\oplus E^c$ has positive distance from $\partial\Omega$, then $\Omega$ is a Runge domain. We also discuss additional classes and constructions of flow invariant Runge domains arising from holomorphic dynamics.
Second, we investigate the global linearization of holomorphic vector fields by automorphisms of \(\mathbb {C}^n\). We prove that a complete holomorphic vector field $V$ on $\mathbb{C}^n$ with a globally attracting fixed point, satisfying certain integrability condition can be globally linearized by an automorphism of $\mathbb{C}^n$. As a corollary we obtain the global linearization of vector fields of the form
$V(z)=Az+O(\|z\|^m)$
near $z= 0$, under certain spectral-gap condition. The conjugating automorphism is obtained as the limit of the family $e^{-tA}X_t$, where $X_t$ is the flow of $V$. Some examples are provided for illustration.