Shearing maps and a Runge map of the unit ball which does not embed into a Loewner chain with range ${\mathbb{C}}^n$

In this paper we study the class of “shearing” holomorphic maps of the unit ball of the form $(z,w)\mapsto (z+g(w),w)$. Besides general properties, we use such maps to construct an example of a normalized univalent map of the ball onto a Runge domain in ${\mathbb{C}}^n$ which however cannot be embedded into a Loewner chain whose range is ${\mathbb{C}}^n$.