Attainable boundary point

A point on the boundary of a domain together with the class of equivalent paths leading from the interior of the domain to that point. Let $\xi$ be a point on the boundary $\partial G$ of a domain $G$ in the complex $z$- plane and let there exist a path described by the equation $z = z (t)$, where the function $z (t)$ is defined and continuous on a certain segment $[ \alpha , \beta ]$, $z (t) \in G$ if $\alpha \leq t < \beta$, $z ( \beta ) = \xi$. One then says that this path leads to the point $\xi$( from the inside of $G$) and defines the attainable boundary point represented by $\xi$. Two paths leading to $\xi$ are said to be equivalent (or, defining the same attainable boundary point) if there exists a third path which also leads to $\xi$ from the inside of $G$ and which has non-empty intersections inside $G$ as close to $\xi$ as one pleases with each of the two paths considered. The totality of a point $\xi \in \partial G$ and the class of equivalent paths leading to $\xi$ from the interior of $G$ is said to be an attainable boundary point of the domain $G$. Not every point $\xi \in \partial G$ represents an attainable boundary point; on the other hand, the same point $\xi \in \partial G$ can represent several, or even an infinite set of different, attainable boundary points.

An attainable boundary point is the unique point of a prime end (cf. Limit elements) of the first kind; a (multi-point) prime end of the second kind contains exactly one attainable boundary point, while prime ends of the third and fourth kinds do not contain attainable boundary points. Each point of the boundary of a Jordan domain is attainable.

References

 [1] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian) [2] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 1;6