Boundary correspondence, principle of

A principle formulated in the following way. One says that the principle of boundary correspondence holds for a mapping $ f $ if the facts that $ f $ is a continuous mapping of the closure $ \overline{G}\; $ of a domain $ G $ onto the closure $ \overline{D}\; $ of a domain $ D $ and $ f $ is a homeomorphism of $ \overline{G}\; \setminus G $ onto $ \overline{D}\; \setminus D $ imply that $ f $ is a topological mapping of $ \overline{G}\; $ onto $ \overline{D}\; $. Thus, the principle of boundary correspondence is in some sense converse to the boundary-correspondence principle (cf. Boundary correspondence (under conformal mapping)).

If $ G $ and $ D $ are plane domains with Euclidean boundaries homeomorphic to a circle and $ D $ is bounded, then the principle of boundary correspondence holds for analytic functions $ f $ on $ G $, i.e. $ f $ is a conformal mapping of $ G $ onto $ D $. In addition to that given above, various other forms of the principle of boundary correspondence are commonly used for conformal mappings (see [1]). The principle of boundary correspondence has been verified for orientable mappings in Euclidean space (see [2]).

References

[1]	M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)
[2]	L.D. Kudryavtsev, "On differentiable mappings" Dokl. Akad. Nauk SSSR , 95 : 5 (1954) pp. 921–923 (In Russian)
[3]	S.I. Pinchuk, "Holomorphic equivalence of certain classes of domains in $\CC^n$" Math. USSR-Sb. , 111 (153) : 1 (180) pp. 67–94; 159 Mat. Sb. , 111 (153) : 1 (1980) pp. 67–94; 159