Prime end
In mathematics, the prime end compactification is a method to compactify a topological disc (i.e. a simply connected open set in the plane) by adding the boundary circle in an appropriate way.
Historical notes
[edit]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.[1] The theory has been generalized to more general open sets.[2] 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.[2] Milnor (2006) gives an accessible introduction to prime ends in the context of complex dynamical systems.
Formal definition
[edit]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.[3]
Applications
[edit]Carathéodory's principal theorem on the correspondence between boundaries under conformal mappings can be expressed as follows:
If ƒ maps the unit disk conformally and one-to-one onto the domain B, it induces a one-to-one mapping between the points on the unit circle and the prime ends of B.
Notes
[edit]- ^ (Epstein 1981, p. 385).
- ^ a b (Epstein 1981, §2).
- ^ A more precise and formal definition of the concepts of "chains of arcs" and of their equivalence classes is given in the references cited.
References
[edit] | This article includes a list of general references, but it lacks sufficient corresponding 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.
- Epstein, D. B. A. (3 May 1981), "Prime Ends", Proceedings of the London Mathematical Society, s3–42 (3), Oxford: Oxford University Press: 385–414, doi:10.1112/plms/s3-42.3.385, MR 0614728, Zbl 0491.30027.
- Milnor, John (2006) [1999], Dynamics in one complex variable, Annals of Mathematics Studies, vol. 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, ISBN 978-0-691-12488-9,
- "Limit elements", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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