Prime end
Encyclopedia
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 map
s 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 expressed as follows:
If ƒ maps the unit 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 equivalent 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 more precise definition of "chains of arc" and their equivalent 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.
Compactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
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
Constantin Carathéodory
Constantin Carathéodory was a Greek mathematician. He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory...
to describe the boundary behavior of conformal map
Conformal map
In mathematics, a conformal map is a function which preserves angles. In the most common case the function is between domains in the complex plane.More formally, a map,...
s in the complex plane
Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
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 expressed as follows:
If ƒ maps the unit conformally and one-to-one onto the domain B, it induces a one-to-one mapping between the points on the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
and the prime ends of B.
The set of prime ends of the domain B is the set of equivalent 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 more precise definition of "chains of arc" and their equivalent 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.