Limit elements

boundary elements, prime ends, of a domain

Elements of a domain $ B $ in the complex plane that are defined as follows. Let $ B $ be a simply-connected domain of the extended complex plane, and let $ \partial B $ be the boundary of $ B $. A section $ c $ of $ B $ is defined as any simple Jordan arc $ c = \overline{ {ab }}\; $, closed in the spherical metric, with ends $ a $ and $ b $( including the cases $ a = b $ and $ b = \infty $), such that $ a, b $ belong to $ \partial B $, and such that the arc $ c $ subdivides $ B $ into two subdomains such that the boundary of each of them contains a point belonging to $ \partial B $ and different from $ a $ and $ b $. A sequence $ K $ of sections $ c _ {n} $ of a domain $ B $ is called a chain if: 1) the diameter of $ c _ {n} $ tends to zero as $ n \rightarrow \infty $; 2) for each $ n $ the intersection $ \overline{c}\; _ {n} \cap \overline{c}\; _ {n + 1 } $ is empty; and 3) any path connecting a fixed point $ 0 \in B $ in $ B $ with the section $ c _ {n} $, $ n > 1 $, intersects $ c _ {n - 1 } $. Two chains $ K = \{ c _ {n} \} $ and $ K ^ \prime = \{ c _ {n} ^ \prime \} $ in $ B $ are equivalent if any section $ c _ {n} $ separates in $ B $ the point $ 0 $ from all sections $ c _ {n} ^ \prime $, except for a finite number of them. An equivalence class of chains in $ B $ is called a limit element, or prime end, of $ B $.

Let $ P $ be a prime end of $ B $ defined by a chain $ K = \{ c _ {n} \} $, and let $ B _ {n} $ be that one of the two subdomains into which $ B $ is subdivided by $ c _ {n} $ which does not contain $ 0 $. The set $ I ( P) = \cap _ {n = 1 } ^ \infty \overline{B}\; _ {n} $ is called the impression or the support of the prime end. The impression of a prime end consists of boundary points and does not depend on the selection of the chain $ K $ in the equivalence class. A principal point of a prime end is a point of it to which sections of at least one of the chains defining the prime end converge. A neighbouring, or subsidiary, point of a prime end is any point of it which is not a principal point of it. Any prime end contains at least one principal point. The principal points of a prime end form a closed set. The following is the Carathéodory classification [1] of prime ends: Elements of the first kind contain a single principal point and no subsidiary points; elements of the second kind contain one principal point and infinitely many subsidiary points; elements of the third kind contain a continuum of principal points and no subsidiary points; elements of the fourth kind contain a continuum of principal points and infinitely many subsidiary points.

Another, equivalent, definition was given by P. Koebe [2]. It is based on equivalence classes of paths. The principal theorem in the theory of prime ends is the theorem of Carathéodory: Under a univalent conformal mapping of a domain $ B $ onto the unit disc $ | \zeta | \leq 1 $ there is a one-to-one correspondence between the points of the circle and the prime ends of $ B $, and each sequence of points of $ B $ which converges to a prime end $ P $ becomes a sequence of points in the unit disc which converge to a point $ \zeta _ {0} $, $ | \zeta _ {0} | = 1 $, this point being the image of $ P $.

References

[1]	C. Carathéodory, "Ueber die Begrenzung einfach zusammenhängender Gebiete" Math. Ann. , 73 (1913) pp. 323–370
[2]	P. Koebe, "Abhandlungen zur Theorie der konformen Abbildung. I" J. Reine Angew. Math. , 145 (1915) pp. 177–223
[3]	G.D. Suvorov, "Families of plane topological mappings" , Novosibirsk (1965) (In Russian)
[4]	A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian)
[5]	E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9

Comments

Instead of "section" the phrase cross cut or cut is also used.

The (only) point of a prime end of the first kind is an accessible boundary point (cf. Attainable boundary point). See also Conformal mapping, boundary properties of a.

Instead of prime end one also finds Carathéodory end in the literature.

There is a second, not entirely dissimilar notion in the literature which goes by the name "end of a topological spaceend" . This refers to the ends of a topological space.

References

[a1]	M. Ohtsuka, "Dirichlet problem, extremal length and prime ends" , v. Nostrand (1967)