Approximation of biholomorphic maps between Runge domains by holomorphic automorphisms

Abstract We show that biholomorphic maps between certain pairs of Runge domains in the complex affine space $${\mathbb {C} }^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , $$n&gt;1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , are limits of holomorphic automorphisms of $${\mathbb {C} }^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . A similar result holds for volume preserving maps and also in Stein manifolds with the density property. This generalises several results in the literature with considerably simpler proofs.