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Difference between revisions of "Attainable boundary arc"

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''of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013890/a0138901.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013890/a0138902.png" />-plane''
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''of a domain $G$ in the $z$-plane''
  
A Jordan arc forming part of the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013890/a0138903.png" /> and at the same time forming part of the boundary of some Jordan domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013890/a0138904.png" />. Each point on an attainable boundary arc is an attainable (from the inside of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013890/a0138905.png" />) boundary point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013890/a0138906.png" /> (cf. [[Attainable boundary point|Attainable boundary point]]). A conformal mapping of a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013890/a0138907.png" /> onto the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013890/a0138908.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013890/a0138909.png" /> can be continuously extended to the non-terminal points of an attainable boundary arc to a homeomorphism of the open attainable boundary arc onto some open arc of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013890/a01389010.png" />.
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A Jordan arc forming part of the boundary of $G$ and at the same time forming part of the boundary of some Jordan domain $g \subset G$. Each point on an attainable boundary arc is an attainable (from the inside of $g$) boundary point of $G$ (cf. [[Attainable boundary point]]). A conformal mapping of a simply-connected domain $G$ onto the unit disc $D = \{ z : |z| < 1 \}$ can be continuously extended to the non-terminal points of an attainable boundary arc to a homeomorphism of the open attainable boundary arc onto some open arc of the circle $|z| = 1$.
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  pp. Chapt. 2  (Translated from Russian)</TD></TR></table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  pp. Chapt. 2  (Translated from Russian)</TD></TR>
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Latest revision as of 19:25, 13 December 2015

of a domain $G$ in the $z$-plane

A Jordan arc forming part of the boundary of $G$ and at the same time forming part of the boundary of some Jordan domain $g \subset G$. Each point on an attainable boundary arc is an attainable (from the inside of $g$) boundary point of $G$ (cf. Attainable boundary point). A conformal mapping of a simply-connected domain $G$ onto the unit disc $D = \{ z : |z| < 1 \}$ can be continuously extended to the non-terminal points of an attainable boundary arc to a homeomorphism of the open attainable boundary arc onto some open arc of the circle $|z| = 1$.


Comments

The standard Western terminology is accessible boundary arc and accessible boundary point, see e.g. [a1].

References

[a1] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9
[a2] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) pp. Chapt. 2 (Translated from Russian)
How to Cite This Entry:
Attainable boundary arc. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Attainable_boundary_arc&oldid=36924
This article was adapted from an original article by E.P. Dolzhenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article