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One of the basic results in the theory of [[Boundary properties of analytic functions|boundary properties of analytic functions]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p0728801.png" /> be a meromorphic function in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p0728802.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p0728803.png" /> be the open angle with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p0728804.png" /> on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p0728805.png" /> formed by two chords of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p0728806.png" /> passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p0728807.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p0728808.png" /> is called a Plessner point (or it is said that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p0728809.png" /> has the Plessner property) if in every arbitrarily small angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288010.png" /> there exists a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288011.png" /> such that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288012.png" /></td> </tr></table>
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for every value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288013.png" /> in the extended complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288014.png" />. The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288015.png" /> is called a Fatou point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288016.png" /> if there exists a single unique limit
+
One of the basic results in the theory of [[Boundary properties of analytic functions|boundary properties of analytic functions]]. Let  $  f( z) $
 +
be a meromorphic function in the unit disc  $  D = \{ {z \in \mathbf C } : {| z | < 1 } \} $
 +
and let  $  \Delta = \Delta ( e ^ {i \theta } ) $
 +
be the open angle with vertex  $  e ^ {i \theta } $
 +
on the circle  $  \Gamma = \{ {z \in \mathbf C } : {| z | = 1 } \} $
 +
formed by two chords of  $  D $
 +
passing through  $  e ^ {i \theta } $.  
 +
The point $  e ^ {i \theta } $
 +
is called a Plessner point (or it is said that  $  e ^ {i \theta } $
 +
has the Plessner property) if in every arbitrarily small angle  $  \Delta $
 +
there exists a sequence  $  \{ z _ {k} \} \subset  \Delta $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288017.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {k \rightarrow \infty }  z _ {k}  = e ^ {i \theta } ,\ \
 +
\lim\limits _ {k \rightarrow \infty }  f( z _ {k} )  = w
 +
$$
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288018.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288019.png" /> within any angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288020.png" />. Plessner's theorem [[#References|[1]]]: Almost-all points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288021.png" /> with respect to the Lebesgue measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288022.png" /> are either Fatou points or Plessner points.
+
for every value  $  w $
 +
in the extended complex plane  $  \overline{\mathbf C}\; $.  
 +
The point  $  e ^ {i \theta } $
 +
is called a Fatou point for  $  f( z) $
 +
if there exists a single unique limit
  
It is also known that the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288023.png" /> of all Plessner points has type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288024.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288025.png" />. Examples have been constructed of analytic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288026.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288027.png" /> is dense on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288028.png" /> and has arbitrary given Lebesgue measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288030.png" /> [[#References|[3]]]. Plessner's theorem applies to any meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288031.png" /> in any simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288032.png" /> with a rectifiable boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288033.png" />. In that case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288034.png" /> is a Fatou point if the following limit exists (cf. also [[Cluster set|Cluster set]]):
+
$$
 +
\lim\limits  f( z) =  A
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288035.png" /></td> </tr></table>
+
as  $  z $
 +
tends to  $  e ^ {i \theta } $
 +
within any angle  $  \Delta $.
 +
Plessner's theorem [[#References|[1]]]: Almost-all points on  $  \Gamma $
 +
with respect to the Lebesgue measure on  $  \Gamma $
 +
are either Fatou points or Plessner points.
  
as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288036.png" /> along any non-tangential path; the definition of a Plessner point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288037.png" /> must be altered in such a way that one considers angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288038.png" /> with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288039.png" /> and sides forming angles less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288040.png" /> with the normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288041.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288042.png" /> [[#References|[2]]].
+
It is also known that the set  $  P( f  ) $
 +
of all Plessner points has type  $  G _  \delta  $
 +
on  $  \Gamma $.
 +
Examples have been constructed of analytic functions in  $  D $
 +
for which  $  P( f  ) $
 +
is dense on  $  \Gamma $
 +
and has arbitrary given Lebesgue measure  $  \mathop{\rm mes}  P( f  ) = m $,
 +
0 \leq  m < 2 \pi $[[#References|[3]]]. Plessner's theorem applies to any meromorphic function  $  f( z) $
 +
in any simply-connected domain  $  D $
 +
with a rectifiable boundary  $  \Gamma $.  
 +
In that case,  $  \zeta \in \Gamma $
 +
is a Fatou point if the following limit exists (cf. also [[Cluster set|Cluster set]]):
 +
 
 +
$$
 +
\lim\limits  f( z)  =  A,\ \
 +
z \in D,
 +
$$
 +
 
 +
as  $  z \rightarrow \zeta $
 +
along any non-tangential path; the definition of a Plessner point $  \zeta \in \Gamma $
 +
must be altered in such a way that one considers angles $  \Delta $
 +
with vertex $  \zeta $
 +
and sides forming angles less than $  \pi /2 $
 +
with the normal to $  \Gamma $
 +
at $  \zeta $[[#References|[2]]].
  
 
Meier's theorem is an analogue of Plessner's theorem in terms of categories of sets (cf. [[Meier theorem|Meier theorem]]).
 
Meier's theorem is an analogue of Plessner's theorem in terms of categories of sets (cf. [[Meier theorem|Meier theorem]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Plessner,  "Über das Verhalten analytischer Funktionen auf dem Rande des Definitionsbereiches"  ''J. Reine Angew. Math.'' , '''158'''  (1928)  pp. 219–227</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.J. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Plessner,  "Über das Verhalten analytischer Funktionen auf dem Rande des Definitionsbereiches"  ''J. Reine Angew. Math.'' , '''158'''  (1928)  pp. 219–227</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. [I.I. Privalov] Priwalow,  "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft.  (1956)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.J. Lohwater,  "The boundary behaviour of analytic functions"  ''Itogi Nauk. i Tekhn. Mat. Anal.'' , '''10'''  (1973)  pp. 99–259  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Angles of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288043.png" /> are called Stolz angles.
+
Angles of the form $  \Delta $
 +
are called Stolz angles.
  
 
A good reference is [[#References|[a3]]], to which [[#References|[3]]] is a Russian sequel.
 
A good reference is [[#References|[a3]]], to which [[#References|[3]]] is a Russian sequel.
  
Plessner's theorem has a complete analogue for the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288044.png" />, cf. [[#References|[a1]]]: Every holomorphic function on the unit ball decomposes the boundary into three measurable sets, as in the classical case.
+
Plessner's theorem has a complete analogue for the unit ball in $  \mathbf C  ^ {n} $,  
 +
cf. [[#References|[a1]]]: Every holomorphic function on the unit ball decomposes the boundary into three measurable sets, as in the classical case.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288045.png" />" , Springer  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Tsuji,  "Potential theory in modern function theory" , Chelsea, reprint  (1975)</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top">  K. Noshiro,  "Cluster sets" , Springer  (1960)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072880/p07288045.png" />" , Springer  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Tsuji,  "Potential theory in modern function theory" , Chelsea, reprint  (1975)</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top">  K. Noshiro,  "Cluster sets" , Springer  (1960)</TD></TR></table>

Latest revision as of 08:06, 6 June 2020


One of the basic results in the theory of boundary properties of analytic functions. Let $ f( z) $ be a meromorphic function in the unit disc $ D = \{ {z \in \mathbf C } : {| z | < 1 } \} $ and let $ \Delta = \Delta ( e ^ {i \theta } ) $ be the open angle with vertex $ e ^ {i \theta } $ on the circle $ \Gamma = \{ {z \in \mathbf C } : {| z | = 1 } \} $ formed by two chords of $ D $ passing through $ e ^ {i \theta } $. The point $ e ^ {i \theta } $ is called a Plessner point (or it is said that $ e ^ {i \theta } $ has the Plessner property) if in every arbitrarily small angle $ \Delta $ there exists a sequence $ \{ z _ {k} \} \subset \Delta $ such that

$$ \lim\limits _ {k \rightarrow \infty } z _ {k} = e ^ {i \theta } ,\ \ \lim\limits _ {k \rightarrow \infty } f( z _ {k} ) = w $$

for every value $ w $ in the extended complex plane $ \overline{\mathbf C}\; $. The point $ e ^ {i \theta } $ is called a Fatou point for $ f( z) $ if there exists a single unique limit

$$ \lim\limits f( z) = A $$

as $ z $ tends to $ e ^ {i \theta } $ within any angle $ \Delta $. Plessner's theorem [1]: Almost-all points on $ \Gamma $ with respect to the Lebesgue measure on $ \Gamma $ are either Fatou points or Plessner points.

It is also known that the set $ P( f ) $ of all Plessner points has type $ G _ \delta $ on $ \Gamma $. Examples have been constructed of analytic functions in $ D $ for which $ P( f ) $ is dense on $ \Gamma $ and has arbitrary given Lebesgue measure $ \mathop{\rm mes} P( f ) = m $, $ 0 \leq m < 2 \pi $[3]. Plessner's theorem applies to any meromorphic function $ f( z) $ in any simply-connected domain $ D $ with a rectifiable boundary $ \Gamma $. In that case, $ \zeta \in \Gamma $ is a Fatou point if the following limit exists (cf. also Cluster set):

$$ \lim\limits f( z) = A,\ \ z \in D, $$

as $ z \rightarrow \zeta $ along any non-tangential path; the definition of a Plessner point $ \zeta \in \Gamma $ must be altered in such a way that one considers angles $ \Delta $ with vertex $ \zeta $ and sides forming angles less than $ \pi /2 $ with the normal to $ \Gamma $ at $ \zeta $[2].

Meier's theorem is an analogue of Plessner's theorem in terms of categories of sets (cf. Meier theorem).

References

[1] A.I. Plessner, "Über das Verhalten analytischer Funktionen auf dem Rande des Definitionsbereiches" J. Reine Angew. Math. , 158 (1928) pp. 219–227
[2] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian)
[3] A.J. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)

Comments

Angles of the form $ \Delta $ are called Stolz angles.

A good reference is [a3], to which [3] is a Russian sequel.

Plessner's theorem has a complete analogue for the unit ball in $ \mathbf C ^ {n} $, cf. [a1]: Every holomorphic function on the unit ball decomposes the boundary into three measurable sets, as in the classical case.

References

[a1] W. Rudin, "Function theory in the unit ball in " , Springer (1981)
[a2] M. Tsuji, "Potential theory in modern function theory" , Chelsea, reprint (1975)
[a3] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9
[a4] K. Noshiro, "Cluster sets" , Springer (1960)
How to Cite This Entry:
Plessner theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Plessner_theorem&oldid=17470
This article was adapted from an original article by E.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article