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 a circle in an appropriate way.

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:

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.

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^[1] 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^[2] for a very accessible introduction in the context of complex dynamical systems.

References

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

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