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''boundary elements, prime ends, of a domain''
 
''boundary elements, prime ends, of a domain''
  
Elements of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l0588601.png" /> in the complex plane that are defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l0588602.png" /> be a simply-connected domain of the extended complex plane, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l0588603.png" /> be the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l0588604.png" />. A section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l0588605.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l0588606.png" /> is defined as any simple Jordan arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l0588607.png" />, closed in the spherical metric, with ends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l0588608.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l0588609.png" /> (including the cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886011.png" />), such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886012.png" /> belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886013.png" />, and such that the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886014.png" /> subdivides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886015.png" /> into two subdomains such that the boundary of each of them contains a point belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886016.png" /> and different from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886018.png" />. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886019.png" /> of sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886020.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886021.png" /> is called a chain if: 1) the diameter of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886022.png" /> tends to zero as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886023.png" />; 2) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886024.png" /> the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886025.png" /> is empty; and 3) any path connecting a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886026.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886027.png" /> with the section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886029.png" />, intersects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886030.png" />. Two chains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886033.png" /> are equivalent if any section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886034.png" /> separates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886035.png" /> the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886036.png" /> from all sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886037.png" />, except for a finite number of them. An equivalence class of chains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886038.png" /> is called a limit element, or prime end, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886039.png" />.
+
Elements of a domain $  B $
 +
in the complex plane that are defined as follows. Let $  B $
 +
be a simply-connected domain of the extended complex plane, and let $  \partial  B $
 +
be the boundary of $  B $.  
 +
A section $  c $
 +
of $  B $
 +
is defined as any simple Jordan arc $  c = \overline{ {ab }}\; $,  
 +
closed in the spherical metric, with ends $  a $
 +
and $  b $(
 +
including the cases $  a = b $
 +
and $  b = \infty $),  
 +
such that $  a, b $
 +
belong to $  \partial  B $,  
 +
and such that the arc $  c $
 +
subdivides $  B $
 +
into two subdomains such that the boundary of each of them contains a point belonging to $  \partial  B $
 +
and different from $  a $
 +
and $  b $.  
 +
A sequence $  K $
 +
of sections $  c _ {n} $
 +
of a domain $  B $
 +
is called a chain if: 1) the diameter of $  c _ {n} $
 +
tends to zero as $  n \rightarrow \infty $;  
 +
2) for each $  n $
 +
the intersection $  \overline{c}\; _ {n} \cap \overline{c}\; _ {n + 1 }  $
 +
is empty; and 3) any path connecting a fixed point 0 \in B $
 +
in $  B $
 +
with the section $  c _ {n} $,  
 +
$  n > 1 $,  
 +
intersects $  c _ {n - 1 }  $.  
 +
Two chains $  K = \{ c _ {n} \} $
 +
and $  K  ^  \prime  = \{ c _ {n}  ^  \prime  \} $
 +
in $  B $
 +
are equivalent if any section $  c _ {n} $
 +
separates in $  B $
 +
the point 0 $
 +
from all sections $  c _ {n}  ^  \prime  $,  
 +
except for a finite number of them. An equivalence class of chains in $  B $
 +
is called a limit element, or prime end, of $  B $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886040.png" /> be a prime end of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886041.png" /> defined by a chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886042.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886043.png" /> be that one of the two subdomains into which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886044.png" /> is subdivided by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886045.png" /> which does not contain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886046.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886047.png" /> is called the impression or the support of the prime end. The impression of a prime end consists of boundary points and does not depend on the selection of the chain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886048.png" /> in the equivalence class. A principal point of a prime end is a point of it to which sections of at least one of the chains defining the prime end converge. A neighbouring, or subsidiary, point of a prime end is any point of it which is not a principal point of it. Any prime end contains at least one principal point. The principal points of a prime end form a closed set. The following is the Carathéodory classification [[#References|[1]]] of prime ends: Elements of the first kind contain a single principal point and no subsidiary points; elements of the second kind contain one principal point and infinitely many subsidiary points; elements of the third kind contain a continuum of principal points and no subsidiary points; elements of the fourth kind contain a continuum of principal points and infinitely many subsidiary points.
+
Let $  P $
 +
be a prime end of $  B $
 +
defined by a chain $  K = \{ c _ {n} \} $,  
 +
and let $  B _ {n} $
 +
be that one of the two subdomains into which $  B $
 +
is subdivided by $  c _ {n} $
 +
which does not contain 0 $.  
 +
The set $  I ( P) = \cap _ {n = 1 }  ^  \infty  \overline{B}\; _ {n} $
 +
is called the impression or the support of the prime end. The impression of a prime end consists of boundary points and does not depend on the selection of the chain $  K $
 +
in the equivalence class. A principal point of a prime end is a point of it to which sections of at least one of the chains defining the prime end converge. A neighbouring, or subsidiary, point of a prime end is any point of it which is not a principal point of it. Any prime end contains at least one principal point. The principal points of a prime end form a closed set. The following is the Carathéodory classification [[#References|[1]]] of prime ends: Elements of the first kind contain a single principal point and no subsidiary points; elements of the second kind contain one principal point and infinitely many subsidiary points; elements of the third kind contain a continuum of principal points and no subsidiary points; elements of the fourth kind contain a continuum of principal points and infinitely many subsidiary points.
  
Another, equivalent, definition was given by P. Koebe [[#References|[2]]]. It is based on equivalence classes of paths. The principal theorem in the theory of prime ends is the theorem of Carathéodory: Under a univalent [[Conformal mapping|conformal mapping]] of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886049.png" /> onto the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886050.png" /> there is a one-to-one correspondence between the points of the circle and the prime ends of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886051.png" />, and each sequence of points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886052.png" /> which converges to a prime end <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886053.png" /> becomes a sequence of points in the unit disc which converge to a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886055.png" />, this point being the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058860/l05886056.png" />.
+
Another, equivalent, definition was given by P. Koebe [[#References|[2]]]. It is based on equivalence classes of paths. The principal theorem in the theory of prime ends is the theorem of Carathéodory: Under a univalent [[Conformal mapping|conformal mapping]] of a domain $  B $
 +
onto the unit disc $  | \zeta | \leq  1 $
 +
there is a one-to-one correspondence between the points of the circle and the prime ends of $  B $,  
 +
and each sequence of points of $  B $
 +
which converges to a prime end $  P $
 +
becomes a sequence of points in the unit disc which converge to a point $  \zeta _ {0} $,  
 +
$  | \zeta _ {0} | = 1 $,  
 +
this point being the image of $  P $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Carathéodory,  "Ueber die Begrenzung einfach zusammenhängender Gebiete"  ''Math. Ann.'' , '''73'''  (1913)  pp. 323–370</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Koebe,  "Abhandlungen zur Theorie der konformen Abbildung. I"  ''J. Reine Angew. Math.'' , '''145'''  (1915)  pp. 177–223</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.D. Suvorov,  "Families of plane topological mappings" , Novosibirsk  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Carathéodory,  "Ueber die Begrenzung einfach zusammenhängender Gebiete"  ''Math. Ann.'' , '''73'''  (1913)  pp. 323–370</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P. Koebe,  "Abhandlungen zur Theorie der konformen Abbildung. I"  ''J. Reine Angew. Math.'' , '''145'''  (1915)  pp. 177–223</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.D. Suvorov,  "Families of plane topological mappings" , Novosibirsk  (1965)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.I. Markushevich,  "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  E.F. Collingwood,  A.J. Lohwater,  "The theory of cluster sets" , Cambridge Univ. Press  (1966)  pp. Chapt. 9</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 22:16, 5 June 2020


boundary elements, prime ends, of a domain

Elements of a domain $ B $ in the complex plane that are defined as follows. Let $ B $ be a simply-connected domain of the extended complex plane, and let $ \partial B $ be the boundary of $ B $. A section $ c $ of $ B $ is defined as any simple Jordan arc $ c = \overline{ {ab }}\; $, closed in the spherical metric, with ends $ a $ and $ b $( including the cases $ a = b $ and $ b = \infty $), such that $ a, b $ belong to $ \partial B $, and such that the arc $ c $ subdivides $ B $ into two subdomains such that the boundary of each of them contains a point belonging to $ \partial B $ and different from $ a $ and $ b $. A sequence $ K $ of sections $ c _ {n} $ of a domain $ B $ is called a chain if: 1) the diameter of $ c _ {n} $ tends to zero as $ n \rightarrow \infty $; 2) for each $ n $ the intersection $ \overline{c}\; _ {n} \cap \overline{c}\; _ {n + 1 } $ is empty; and 3) any path connecting a fixed point $ 0 \in B $ in $ B $ with the section $ c _ {n} $, $ n > 1 $, intersects $ c _ {n - 1 } $. Two chains $ K = \{ c _ {n} \} $ and $ K ^ \prime = \{ c _ {n} ^ \prime \} $ in $ B $ are equivalent if any section $ c _ {n} $ separates in $ B $ the point $ 0 $ from all sections $ c _ {n} ^ \prime $, except for a finite number of them. An equivalence class of chains in $ B $ is called a limit element, or prime end, of $ B $.

Let $ P $ be a prime end of $ B $ defined by a chain $ K = \{ c _ {n} \} $, and let $ B _ {n} $ be that one of the two subdomains into which $ B $ is subdivided by $ c _ {n} $ which does not contain $ 0 $. The set $ I ( P) = \cap _ {n = 1 } ^ \infty \overline{B}\; _ {n} $ is called the impression or the support of the prime end. The impression of a prime end consists of boundary points and does not depend on the selection of the chain $ K $ in the equivalence class. A principal point of a prime end is a point of it to which sections of at least one of the chains defining the prime end converge. A neighbouring, or subsidiary, point of a prime end is any point of it which is not a principal point of it. Any prime end contains at least one principal point. The principal points of a prime end form a closed set. The following is the Carathéodory classification [1] of prime ends: Elements of the first kind contain a single principal point and no subsidiary points; elements of the second kind contain one principal point and infinitely many subsidiary points; elements of the third kind contain a continuum of principal points and no subsidiary points; elements of the fourth kind contain a continuum of principal points and infinitely many subsidiary points.

Another, equivalent, definition was given by P. Koebe [2]. It is based on equivalence classes of paths. The principal theorem in the theory of prime ends is the theorem of Carathéodory: Under a univalent conformal mapping of a domain $ B $ onto the unit disc $ | \zeta | \leq 1 $ there is a one-to-one correspondence between the points of the circle and the prime ends of $ B $, and each sequence of points of $ B $ which converges to a prime end $ P $ becomes a sequence of points in the unit disc which converge to a point $ \zeta _ {0} $, $ | \zeta _ {0} | = 1 $, this point being the image of $ P $.

References

[1] C. Carathéodory, "Ueber die Begrenzung einfach zusammenhängender Gebiete" Math. Ann. , 73 (1913) pp. 323–370
[2] P. Koebe, "Abhandlungen zur Theorie der konformen Abbildung. I" J. Reine Angew. Math. , 145 (1915) pp. 177–223
[3] G.D. Suvorov, "Families of plane topological mappings" , Novosibirsk (1965) (In Russian)
[4] A.I. Markushevich, "Theory of functions of a complex variable" , 3 , Chelsea (1977) (Translated from Russian)
[5] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9

Comments

Instead of "section" the phrase cross cut or cut is also used.

The (only) point of a prime end of the first kind is an accessible boundary point (cf. Attainable boundary point). See also Conformal mapping, boundary properties of a.

Instead of prime end one also finds Carathéodory end in the literature.

There is a second, not entirely dissimilar notion in the literature which goes by the name "end of a topological spaceend" . This refers to the ends of a topological space.

References

[a1] M. Ohtsuka, "Dirichlet problem, extremal length and prime ends" , v. Nostrand (1967)
How to Cite This Entry:
Limit elements. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Limit_elements&oldid=47638
This article was adapted from an original article by E.G. Goluzina (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article