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, too.
Carathéodory's principal theorem on the correspondence between boundaries under conformal mappings can be expressed as follows:
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. (for a more precise definition of "chains of arcs" and their equivalence classes see the references)
The expository paper of Epstein provides a good account of this theory with complete proofs. It also introduces a definition which make sense in any open set and dimension. See also Milnor for a very accessible introduction in the context of complex dynamical systems.
||This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. (May 2010)|
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.
- Hazewinkel, Michiel, ed. (2001), "Limit elements", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- D. B. A. Epstein, Prime Ends, Proc. London Math. Soc. 1981 s3-42: 385-414. Epstein, D. B. A. (1981). "Prime Ends". Proceedings of the London Mathematical Society (3): 385–414. doi:10.1112/plms/s3-42.3.385.
- J. Milnor, Dynamics in one complex variable. Third edition. Annals of Mathematics Studies, 160. Princeton University Press, Princeton, NJ, 2006. ISBN 978-0-691-12488-9; 0-691-12488-4
|This topology-related article is a stub. You can help Wikipedia by expanding it.|