# 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.

## Historical notes

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]} Also Milnor (2006) gives an accessible introduction to prime ends in the context of complex dynamical systems.

## Formal definition

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

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

- ↑ (Epstein 1981, p. 385).
- 1 2 (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

*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) [1999],
*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, ISBN 978-1-55608-010-4