The concept of prime ends was introduced by Constantin Carathéodory to describe the boundary behavior of conformal maps in the complex plane in geometric terms. The theory has been generalized to more general open sets. The expository paper of Epstein (1981) provides a good account of this theory with complete proofs: it also introduces a definition which make sense in any open set and dimension. Milnor (2006) gives an accessible introduction to prime ends in the context of complex dynamical systems.
The set of prime ends of the domain B is the set of equivalence classes of chains of arcs converging to a point on the boundary of B.
In this way, a point in the boundary may correspond to many points in the prime ends of B, and conversely, many points in the boundary may correspond to a point in the prime ends of B.
Carathéodory's principal theorem on the correspondence between boundaries under conformal mappings can be expressed as follows:
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This article incorporates material from the Citizendium article "Prime ends", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.
- Epstein, D. B. A. (3 May 1981), "Prime Ends", Proceedings of the London Mathematical Society, Oxford: Oxford University Press, s3–42 (3): 385–414, doi:10.1112/plms/s3-42.3.385, MR 0614728, Zbl 0491.30027, (Subscription required (help)).
- John, Milnor (2006) , Dynamics in one complex variable, Annals of Mathematics Studies, 160 (3rd ed.), Princeton, NJ: Princeton University Press, pp. viii+304, doi:10.1515/9781400835539, ISBN 0-691-12488-4, MR 2193309, Zbl 1281.37001 – via De Gruyter, (Subscription required (help)), ISBN 978-0-691-12488-9,
- Hazewinkel, Michiel, ed. (2001) , "Limit elements", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
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