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In this paper we study the class of "shearing" holomorphic maps of the unitball of the form $(z,w)\mapsto (z+g(w), w)$. Besides general properties, we usesuch maps to construct an example of a normalized univalent map of the ballonto a Runge domain in $\mathbb C^n$ which however cannot be embedded into aLoewner chain whose range is $\mathbb C^n$.

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