Prime end

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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[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] Also 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]

  1. ^ (Epstein 1981, p. 385).
  2. ^ a b (Epstein 1981, §2).
  3. ^ 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 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.