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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c0226601.png" /> of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c0226602.png" />, defined on a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c0226603.png" /> with values in the Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c0226604.png" />, at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c0226605.png" /> with respect to a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c0226606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c0226607.png" />''
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$#A+1 = 329 n = 2
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$#C+1 = 329 : ~/encyclopedia/old_files/data/C022/C.0202660 Cluster set
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The set of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c0226608.png" /> for which there exists a sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c0226609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266011.png" />, such that
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{{TEX|auto}}
 +
{{TEX|done}}
  
<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/c/c022/c022660/c02266012.png" /></td> </tr></table>
+
'' $  C ( f, z _ {0} ; S) $
 +
of a function  $  f : G \rightarrow \Omega $,
 +
defined on a domain  $  G \subset  \mathbf C $
 +
with values in the Riemann sphere  $  \Omega $,
 +
at a point  $  z _ {0} \in \overline{G}\; $
 +
with respect to a set  $  S \subset  G $,
 +
$  z _ {0} \in \overline{S}\; $''
  
Every number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266013.png" /> is called a cluster value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266014.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266015.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266016.png" />. The theory of cluster sets is a branch of function theory in which boundary properties of functions are studied in terms of topological and metric properties of various cluster sets.
+
The set of values  $  a \in \Omega $
 +
for which there exists a sequence of points  $  \{ z _ {n} \} _ {n = 1 }  ^  \infty  $,
 +
$  z _ {n} \in S $,
 +
$  \lim\limits _ {n \rightarrow \infty }  z _ {n} = z _ {0} $,
 +
such that
  
If the entire domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266017.png" /> is taken for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266018.png" />, one obtains the full cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266019.png" />; if the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266020.png" /> is strict, the corresponding set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266021.png" /> is sometimes called a partial cluster set. A full cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266022.png" /> is closed; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266023.png" /> is continuous on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266024.png" /> that is locally connected at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266025.png" />, then the cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266026.png" /> is either degenerate, i.e. consists of a single point, or is a non-degenerate continuum. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266027.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266028.png" />, then it is called a total cluster set. A number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266029.png" /> belongs to the set of recurrent values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266031.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266032.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266033.png" /> if there is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266034.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266036.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266037.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266039.png" />. One always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266040.png" />. If for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266041.png" /> there is a path <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266042.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266044.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266045.png" /> ending at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266047.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266048.png" />, and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266049.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266050.png" /> is called an [[Asymptotic value|asymptotic value]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266051.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266052.png" /> (along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266053.png" />). The asymptotic set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266054.png" /> is the set of all asymptotic values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266055.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266056.png" />.
+
$$
 +
\lim\limits _ {n \rightarrow \infty } \
 +
f ( z _ {n} ) = a.
 +
$$
  
The notion of a cluster set was clearly formulated for the first time by P. Painlevé in 1895 (he called it the "region of indeterminacy" , cf. [[#References|[1]]]) in connection with studying an analytic function near one of its singular points and with classifying singularities of such functions. At that time one basically studied three, geometrically most simple, cases in the theory of cluster sets: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266057.png" /> is an isolated point of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266058.png" /> or an interior point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266059.png" />; b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266060.png" /> is the unit disc or, in general, a Jordan domain, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266061.png" /> is a point on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266062.png" />; and c) the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266063.png" /> is an everywhere-discontinuous compactum in the plane (i.e. a totally-disconnected compact set) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266064.png" />. A number of classical results in complex function theory have a formulation in terms of cluster sets. E.g., the [[Sokhotskii theorem|Sokhotskii theorem]], in a somewhat stronger form, states: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266065.png" /> is an isolated point of an everywhere-discontinuous compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266067.png" /> is a meromorphic function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266068.png" />, then the cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266069.png" /> is either degenerate or total. The [[Picard theorem|Picard theorem]], supplementing it, states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266070.png" /> is total, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266071.png" /> is an essential singular point, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266072.png" /> contains at most two different values. Also, in this case
+
Every number  $  a \in C ( f, z _ {0} ; S) $
 +
is called a cluster value of $  f $
 +
at  $  z _ {0} $
 +
with respect to  $  S $.  
 +
The theory of cluster sets is a branch of function theory in which boundary properties of functions are studied in terms of topological and metric properties of various cluster sets.
  
<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/c/c022/c022660/c02266073.png" /></td> </tr></table>
+
If the entire domain  $  G $
 +
is taken for  $  S $,
 +
one obtains the full cluster set  $  C ( f, z _ {0} ;  G) = C ( f, z _ {0} ) $;
 +
if the inclusion  $  S \subset  G $
 +
is strict, the corresponding set  $  C ( f, z _ {0} ;  S) $
 +
is sometimes called a partial cluster set. A full cluster set  $  C ( f, z _ {0} ) $
 +
is closed; if  $  f $
 +
is continuous on a set  $  S $
 +
that is locally connected at  $  z _ {0} \in \overline{S}\; $,
 +
then the cluster set  $  C ( f, z _ {0} ;  S) $
 +
is either degenerate, i.e. consists of a single point, or is a non-degenerate continuum. If  $  C ( f, z _ {0} ;  S) $
 +
coincides with  $  \Omega $,
 +
then it is called a total cluster set. A number  $  a \in \Omega $
 +
belongs to the set of recurrent values  $  R ( f, z _ {0} ;  S) $
 +
of  $  f $
 +
at  $  z _ {0} $
 +
with respect to  $  S $
 +
if there is a sequence  $  \{ z _ {n} \} $
 +
of points  $  z _ {n} \in S $,
 +
$  n = 1, 2 \dots $
 +
$  \lim\limits _ {n \rightarrow \infty }  z _ {n} = z _ {0} $,
 +
such that  $  a = f ( z _ {n} ) $,
 +
$  n = 1, 2 , .  .  . $.
 +
One always has  $  R ( f, z _ {0} ;  S) \subset  C ( f, z _ {0} ;  S) $.
 +
If for some  $  a \in \Omega $
 +
there is a path  $  L $:  
 +
$  z = z ( t) $,
 +
$  0 \leq  t < 1 $,
 +
in  $  G $
 +
ending at a point  $  z _ {0} $,
 +
$  z _ {0} \in \overline{G}\; $,
 +
$  \lim\limits _ {t \rightarrow 1 }  z ( t) = z _ {0} $,
 +
and such that  $  \lim\limits _ {t \rightarrow 1 }  f ( z ( t)) = a $,
 +
then  $  a $
 +
is called an [[Asymptotic value|asymptotic value]] of  $  f $
 +
at  $  z _ {0} $(
 +
along  $  L $).
 +
The asymptotic set  $  A ( f, z _ {0} ;  G) $
 +
is the set of all asymptotic values of  $  f $
 +
at  $  z _ {0} $.
 +
 
 +
The notion of a cluster set was clearly formulated for the first time by P. Painlevé in 1895 (he called it the "region of indeterminacy" , cf. [[#References|[1]]]) in connection with studying an analytic function near one of its singular points and with classifying singularities of such functions. At that time one basically studied three, geometrically most simple, cases in the theory of cluster sets: a)  $  z _ {0} $
 +
is an isolated point of the boundary  $  \partial  G $
 +
or an interior point of  $  G $;  
 +
b)  $  G = D = \{ {z } : {| z | < 1 } \} $
 +
is the unit disc or, in general, a Jordan domain, and  $  z _ {0} $
 +
is a point on the boundary  $  \Gamma = \partial  D $;
 +
and c) the boundary  $  E = \partial  G $
 +
is an everywhere-discontinuous compactum in the plane (i.e. a totally-disconnected compact set) and  $  z _ {0} \in E $.
 +
A number of classical results in complex function theory have a formulation in terms of cluster sets. E.g., the [[Sokhotskii theorem|Sokhotskii theorem]], in a somewhat stronger form, states: If  $  z _ {0} $
 +
is an isolated point of an everywhere-discontinuous compactum  $  E \subset  G $
 +
and  $  f $
 +
is a meromorphic function on  $  G \setminus  E $,
 +
then the cluster set  $  C ( f, z _ {0} ;  G \setminus  E) $
 +
is either degenerate or total. The [[Picard theorem|Picard theorem]], supplementing it, states that if  $  C ( f, z _ {0} ;  G \setminus  E) $
 +
is total, i.e. if  $  z _ {0} $
 +
is an essential singular point, then the set  $  CR ( f, z _ {0} ;  G \setminus  E) = \Omega \setminus  R ( f, z _ {0} ;  G \setminus  E) $
 +
contains at most two different values. Also, in this case
 +
 
 +
$$
 +
CR ( f, z _ {0} ;  G \setminus  E)  \subset  \
 +
A ( f, z _ {0} ;  G \setminus  E)
 +
$$
  
 
(the [[Iversen theorem|Iversen theorem]]).
 
(the [[Iversen theorem|Iversen theorem]]).
  
The main result related to the theory of the behaviour of meromorphic functions near "thin" boundaries (the Painlevé theory) is (cf. [[#References|[1]]], [[#References|[2]]]): If a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266074.png" /> has linear Hausdorff measure zero, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266075.png" />, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266076.png" /> is meromorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266077.png" />, then for every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266078.png" /> the cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266079.png" /> is either degenerate or total; moreover, in the first case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266080.png" /> is also meromorphic at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266081.png" />. Thus, a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266082.png" /> for which the cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266083.png" /> is degenerate is a removable singular point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266084.png" />; the study of removable sets of various function classes can be regarded as a branch of the theory of cluster sets.
+
The main result related to the theory of the behaviour of meromorphic functions near "thin" boundaries (the Painlevé theory) is (cf. [[#References|[1]]], [[#References|[2]]]): If a set $  E \subset  G $
 +
has linear Hausdorff measure zero, $  \mu ( E) = \mu _ {1} ( E) = 0 $,  
 +
and the function $  f $
 +
is meromorphic in $  G \setminus  E $,  
 +
then for every point $  z _ {0} \in E $
 +
the cluster set $  C ( f, z _ {0} ;  G \setminus  E) $
 +
is either degenerate or total; moreover, in the first case $  f $
 +
is also meromorphic at $  z _ {0} $.  
 +
Thus, a point $  z _ {0} \in E $
 +
for which the cluster set $  C ( f, z _ {0} ;  G \setminus  E) $
 +
is degenerate is a removable singular point of $  f $;  
 +
the study of removable sets of various function classes can be regarded as a branch of the theory of cluster sets.
  
Golubev's theorem is an important strengthening of the theorem of Picard: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266086.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266087.png" /> is meromorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266088.png" />, then the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266089.png" /> has [[Analytic capacity|analytic capacity]] zero at every essential singular point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266090.png" /> (hence its plane measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266091.png" />).
+
Golubev's theorem is an important strengthening of the theorem of Picard: If $  E \subset  G $,  
 +
$  \mu ( E) = 0 $
 +
and $  f $
 +
is meromorphic in $  G \setminus  E $,  
 +
then the set $  CR ( f, z _ {0} ;  G \setminus  E) $
 +
has [[Analytic capacity|analytic capacity]] zero at every essential singular point $  z _ {0} \in E $(
 +
hence its plane measure $  \mu _ {2} ( CR) = 0 $).
  
The work of P. Fatou (1906) on boundary values of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266092.png" /> holomorphic in the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266093.png" /> was the starting point for the theory of cluster sets in the case of continuous boundaries. If such a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266094.png" /> is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266095.png" />, then almost-everywhere (in the sense of the Lebesgue measure) on the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266096.png" /> it has radial and angular (non-tangential) boundary values (Fatou's theorem). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266097.png" /> be an arbitrary point; denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266098.png" /> the chord of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c02266099.png" /> ending at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660100.png" /> and forming with the radius at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660101.png" /> an angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660103.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660104.png" /> be the angular domain with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660105.png" />, consisting of those points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660106.png" /> lying between the chords
+
The work of P. Fatou (1906) on boundary values of functions $  f ( z) $
 +
holomorphic in the unit disc $  D = \{ {z } : {| z | < 1 } \} $
 +
was the starting point for the theory of cluster sets in the case of continuous boundaries. If such a function $  f $
 +
is bounded in $  D $,  
 +
then almost-everywhere (in the sense of the Lebesgue measure) on the circle $  \Gamma = \{ {z } : {| z | = 1 } \} $
 +
it has radial and angular (non-tangential) boundary values (Fatou's theorem). Let $  \zeta = e ^ {i \theta } \in \Gamma $
 +
be an arbitrary point; denote by $  h ( \zeta , \phi ) $
 +
the chord of $  D $
 +
ending at $  \zeta $
 +
and forming with the radius at $  \zeta $
 +
an angle $  \phi $,
 +
$  - \pi /2 < \phi < \pi /2 $.  
 +
Let $  \Delta ( \zeta , \phi _ {1} , \phi _ {2} ) $
 +
be the angular domain with vertex $  \zeta \in \Gamma $,  
 +
consisting of those points of $  D $
 +
lying between the chords
  
<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/c/c022/c022660/c022660107.png" /></td> </tr></table>
+
$$
 +
h ( \zeta , \phi _ {1} ) \ \
 +
\textrm{ and } \ \
 +
h ( \zeta , \phi _ {2} ),\ \
 +
{-
 +
\frac \pi {2}
 +
} < \phi _ {1} <
 +
\phi _ {2} < {
 +
\frac \pi {2}
 +
} .
 +
$$
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660108.png" /> is called a Fatou point, and belongs to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660109.png" />, if the union
+
A point $  \zeta \in \Gamma $
 +
is called a Fatou point, and belongs to the set $  F ( f  ) $,  
 +
if the union
  
<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/c/c022/c022660/c022660110.png" /></td> </tr></table>
+
$$
 +
\cup C ( f, \zeta ; \Delta
 +
( \zeta , \phi _ {1} , \phi _ {2} ))
 +
$$
  
over all angular domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660111.png" /> consists of a single value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660112.png" />, which is called the angular boundary value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660113.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660114.png" />. Another formulation of Fatou's theorem: For a bounded holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660115.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660116.png" /> the decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660118.png" />, holds. This result is supplemented by the F. and M. Riesz uniqueness theorem (1916): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660119.png" /> is holomorphic and bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660120.png" /> and if on some set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660121.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660122.png" />, it has angular boundary values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660123.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660124.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660125.png" />. This statement was proved, independently, by N.N. Luzin and I.I. Privalov (1919), who obtained an essential generalization of it to the case of arbitrary meromorphic functions. In the same year they published a boundary uniqueness theorem for the case of radial boundary values: If a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660126.png" />, holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660127.png" />, has the same radial boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660128.png" /> on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660129.png" /> of the second category and metrically dense on some arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660130.png" />, i.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660132.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660133.png" />.
+
over all angular domains $  \Delta ( \zeta , \phi _ {1} , \phi _ {2} ) $
 +
consists of a single value $  f ( e ^ {i \theta } ) $,  
 +
which is called the angular boundary value of $  f $
 +
at $  \zeta $.  
 +
Another formulation of Fatou's theorem: For a bounded holomorphic function $  f $
 +
in $  D $
 +
the decomposition $  \Gamma = F ( f  ) \cup E $,  
 +
$  \mathop{\rm mes}  E = 0 $,  
 +
holds. This result is supplemented by the F. and M. Riesz uniqueness theorem (1916): If $  f $
 +
is holomorphic and bounded in $  D $
 +
and if on some set $  M \subset  F ( f  ) $,
 +
$  \mathop{\rm mes}  M > 0 $,  
 +
it has angular boundary values $  f ( \zeta ) = a $,  
 +
$  \zeta \in M $,  
 +
then $  f ( z) \equiv a $.  
 +
This statement was proved, independently, by N.N. Luzin and I.I. Privalov (1919), who obtained an essential generalization of it to the case of arbitrary meromorphic functions. In the same year they published a boundary uniqueness theorem for the case of radial boundary values: If a function $  f $,  
 +
holomorphic in $  D $,  
 +
has the same radial boundary value $  a \in \Omega $
 +
on a set $  M $
 +
of the second category and metrically dense on some arc $  \gamma \subset  \Gamma $,  
 +
i.e. if $  \lim\limits _ {r \rightarrow 1 }  f ( re ^ {i \theta } ) = a $,  
 +
$  e ^ {i \theta } \in M $,  
 +
then $  f ( z) \equiv a $.
  
Privalov, in 1936, noted that the statement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660134.png" /> remains true also when the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660135.png" /> are not necessarily equal at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660136.png" />, but belong to a set of (logarithmic) capacity zero. The basic idea and the elements of the proof of the Luzin–Privalov theorem are applicable in the general case of continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660137.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660138.png" />, which was subsequently used in many papers.
+
Privalov, in 1936, noted that the statement $  f ( z) \equiv \textrm{ const } $
 +
remains true also when the values $  a _  \zeta  = \lim\limits _ {r \rightarrow 1 }  f ( re ^ {i \theta } ) $
 +
are not necessarily equal at the points $  \zeta = e ^ {i \theta } \in M $,  
 +
but belong to a set of (logarithmic) capacity zero. The basic idea and the elements of the proof of the Luzin–Privalov theorem are applicable in the general case of continuous mappings $  f $
 +
of $  D $,  
 +
which was subsequently used in many papers.
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660139.png" /> is called a Plessner point, and belongs to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660140.png" />, if the intersection
+
A point $  \zeta \in \Gamma = \{ {z } : {| z | = 1 } \} $
 +
is called a Plessner point, and belongs to the set $  I ( f  ) $,  
 +
if the intersection
  
<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/c/c022/c022660/c022660141.png" /></td> </tr></table>
+
$$
 +
\cap C ( f, \zeta ; \Delta
 +
( \zeta , \phi _ {1} , \phi _ {2} ))
 +
$$
  
over all angular domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660142.png" /> with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660143.png" /> coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660144.png" />. A.I. Plessner proved (1927) that for a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660145.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660146.png" /> almost-all points of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660147.png" /> belong either to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660148.png" /> or to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660149.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660150.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660151.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660152.png" /> is called a Meier point, and belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660153.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660154.png" /> and if the intersection of the chordal cluster sets, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660155.png" />, over all chords drawn at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660156.png" />, coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660157.png" />. K. Meier established (1961) the following analogue of Plessner's theorem in terms of Baire categories: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660158.png" /> is meromorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660159.png" />, then all points of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660160.png" />, with the possible exception of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660161.png" /> of the first category, belong to the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660162.png" />. A more precise statement of Meier's theorem has been obtained, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660163.png" /> is a set of the first category and of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660164.png" /> (cf. [[#References|[12]]]–[[#References|[14]]], in which generalizations of Plessner's and Meier's theorems have been obtained, and in which a converse of Meier's theorem and a characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660165.png" /> have been given).
+
over all angular domains $  \Delta ( \zeta , \phi _ {1} , \phi _ {2} ) $
 +
with vertex $  \zeta $
 +
coincides with $  \Omega $.  
 +
A.I. Plessner proved (1927) that for a meromorphic function $  f $
 +
in $  D $
 +
almost-all points of the boundary $  \Gamma $
 +
belong either to $  F ( f  ) $
 +
or to $  I ( f  ) $,  
 +
i.e. $  \Gamma = F ( f  ) \cup I ( f  ) \cup E $,  
 +
$  \mathop{\rm mes}  E = 0 $.  
 +
A point $  \zeta \in \Gamma $
 +
is called a Meier point, and belongs to $  M ( f  ) $,  
 +
if $  C ( f, \zeta ;  D) \neq \Omega $
 +
and if the intersection of the chordal cluster sets, $  \cap C ( f, \zeta ;  h ( \zeta , \phi )) $,  
 +
over all chords drawn at $  \zeta $,  
 +
coincides with $  C ( f, \zeta ;  D) $.  
 +
K. Meier established (1961) the following analogue of Plessner's theorem in terms of Baire categories: If $  f $
 +
is meromorphic in $  D $,  
 +
then all points of the boundary $  \Gamma $,  
 +
with the possible exception of a set $  E $
 +
of the first category, belong to the union $  M ( f  ) \cup I ( f  ) $.  
 +
A more precise statement of Meier's theorem has been obtained, in which $  E $
 +
is a set of the first category and of type $  F _  \sigma  $(
 +
cf. [[#References|[12]]]–[[#References|[14]]], in which generalizations of Plessner's and Meier's theorems have been obtained, and in which a converse of Meier's theorem and a characterization of $  M ( f  ) $
 +
have been given).
  
 
The work of Fatou served as an original source for the development of fundamental research on boundary properties of analytic functions. The studies of F. and M. Riesz, Luzin, Privalov, R. Nevanlinna, Plessner, V.I. Smirnov, and others were conducted independently of the ideas of Painlevé, and the use of methods related to measure and integration theory, including the notion of Baire categories, is characteristic for them (cf. [[#References|[4]]]–[[#References|[9]]]).
 
The work of Fatou served as an original source for the development of fundamental research on boundary properties of analytic functions. The studies of F. and M. Riesz, Luzin, Privalov, R. Nevanlinna, Plessner, V.I. Smirnov, and others were conducted independently of the ideas of Painlevé, and the use of methods related to measure and integration theory, including the notion of Baire categories, is characteristic for them (cf. [[#References|[4]]]–[[#References|[9]]]).
  
The basic objects of study for F. Iversen and W. Gross were meromorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660166.png" /> in domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660167.png" /> with a Jordan boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660168.png" />. At an arbitrary point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660169.png" />, the boundary cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660170.png" /> is defined as follows: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660171.png" /> denotes the closure of the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660172.png" /> over all points
+
The basic objects of study for F. Iversen and W. Gross were meromorphic functions $  f $
 +
in domains $  D $
 +
with a Jordan boundary $  \Gamma = \partial  D $.  
 +
At an arbitrary point $  \zeta _ {0} \in \Gamma $,  
 +
the boundary cluster set $  C ( f, \zeta _ {0} ;  \Gamma ) $
 +
is defined as follows: If $  M _ {r} $
 +
denotes the closure of the union $  \cup C ( f, \zeta ;  D) $
 +
over all points
  
<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/c/c022/c022660/c022660173.png" /></td> </tr></table>
+
$$
 +
\zeta  \in \
 +
( \Gamma \setminus
 +
\{ \zeta _ {0} \} )
 +
\cap \{ {z } : {
 +
| z - \zeta _ {0} |
 +
< r } \}
 +
,
 +
$$
  
then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660174.png" />. One of the main theorems obtained, independently, by them asserts that, under the conditions stated, the set
+
then $  C ( f, \zeta _ {0} ;  \Gamma ) = \cap _ {r > 0 }  M _ {r} $.  
 +
One of the main theorems obtained, independently, by them asserts that, under the conditions stated, the set
  
<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/c/c022/c022660/c022660175.png" /></td> </tr></table>
+
$$
 +
C _ {i} ( f,\
 +
\zeta _ {0} ; D)  = \
 +
C ( f, \zeta _ {0} ; D) \setminus
 +
C ( f, \zeta _ {0} ; \Gamma )
 +
$$
  
is open (for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660176.png" />), and all values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660177.png" />, with possibly two exceptions, belong to the set of recurrent values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660178.png" />. Moreover, every exceptional value (if existing) is an asymptotic value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660179.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660180.png" />.
+
is open (for any $  \zeta _ {0} \in \Gamma $),  
 +
and all values $  a \in C _ {i} ( f, \zeta _ {0} ;  D) $,  
 +
with possibly two exceptions, belong to the set of recurrent values $  R ( f, \zeta _ {0} ;  D) $.  
 +
Moreover, every exceptional value (if existing) is an asymptotic value of $  f $
 +
at $  \zeta _ {0} $.
  
The research of Iversen and Gross obtained a further development in the work of A. Beurling, W. Seidel (who in 1932 also introduced the term "cluster set" ) and others (cf. [[#References|[5]]]–[[#References|[9]]]). They basically considered the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660181.png" /> belongs to a "small"  set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660182.png" /> on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660183.png" />, having zero linear measure or zero capacity, and studied the cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660184.png" />, defined analogously to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660185.png" />. Methods of potential theory are also used in these studies.
+
The research of Iversen and Gross obtained a further development in the work of A. Beurling, W. Seidel (who in 1932 also introduced the term "cluster set" ) and others (cf. [[#References|[5]]]–[[#References|[9]]]). They basically considered the case when $  \zeta _ {0} $
 +
belongs to a "small" set $  E $
 +
on the boundary $  \Gamma $,  
 +
having zero linear measure or zero capacity, and studied the cluster set $  C ( f, \zeta _ {0} ;  \Gamma \setminus  E) $,  
 +
defined analogously to $  C ( f, \zeta _ {0} ;  \Gamma ) $.  
 +
Methods of potential theory are also used in these studies.
  
The most recent results in this direction are stated below for the case of the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660186.png" />. Suppose a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660187.png" /> on an arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660188.png" /> of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660189.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660190.png" /> having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660191.png" /> is fixed, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660192.png" />. To every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660193.png" /> one assigns a Jordan arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660194.png" /> ending at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660195.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660196.png" /> be the closure of the union <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660197.png" /> over all points
+
The most recent results in this direction are stated below for the case of the disc $  D = \{ {z } : {| z | < 1 } \} $.  
 +
Suppose a set $  E $
 +
on an arc $  \gamma $
 +
of the boundary $  \Gamma $
 +
of $  D $
 +
having $  \mathop{\rm mes}  E = 0 $
 +
is fixed, and let $  \zeta _ {0} \in E $.  
 +
To every point $  \zeta \in \gamma \setminus  E $
 +
one assigns a Jordan arc $  \Lambda _  \zeta  \subset  D $
 +
ending at $  \zeta $.  
 +
Let $  M _ {r}  ^ {*} $
 +
be the closure of the union $  \cup C ( f, \zeta ;  \Lambda _  \zeta  ) $
 +
over all points
  
<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/c/c022/c022660/c022660198.png" /></td> </tr></table>
+
$$
 +
\zeta  \in \
 +
( \gamma \setminus  E)
 +
\cap \{ {z } : {
 +
| z - \zeta _ {0} | < r } \}
 +
$$
  
 
and suppose
 
and suppose
  
<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/c/c022/c022660/c022660199.png" /></td> </tr></table>
+
$$
 +
C  ^ {*} ( f, \zeta _ {0} ; \
 +
\Gamma \setminus  E)  = \
 +
\cap _ {r > 0 }
 +
M _ {r}  ^ {*} .
 +
$$
  
 
Then the set
 
Then the set
  
<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/c/c022/c022660/c022660200.png" /></td> </tr></table>
+
$$
 +
S ( \zeta _ {0} )  = \
 +
C ( f, \zeta _ {0} ;  D) \setminus
 +
C  ^ {*} ( f, \zeta _ {0} ; \
 +
\Gamma \setminus  E)
 +
$$
 +
 
 +
is open, the set  $  S ( \zeta _ {0} ) \setminus  R ( f, \zeta _ {0} ; D) $
 +
has capacity zero, and every value  $  a \in S ( \zeta _ {0} ) \setminus  R ( f, \zeta _ {0} ; D) $
 +
is an asymptotic value of  $  f $
 +
either at  $  \zeta _ {0} $
 +
or at every point of some sequence  $  \{ \zeta _ {n} \} $,
 +
$  \zeta _ {n} \in \Gamma $,
 +
$  n = 1, 2 \dots $
 +
$  \lim\limits _ {n \rightarrow \infty }  \zeta _ {n} = \zeta _ {0} $.
 +
If  $  E $
 +
has capacity zero, then for every connected component  $  S _ {k} ( \zeta _ {0} ) $,
 +
$  k = 1, 2 \dots $
 +
of  $  S ( \zeta _ {0} ) $
 +
the set  $  S _ {k} ( \zeta _ {0} ) \setminus  R ( f, \zeta _ {0} ;  D) $
 +
consists of at most two distinct values.
 +
 
 +
Lindelöf's theorem has been proved using normal families (cf. [[Normal family|Normal family]]): If a holomorphic function  $  f $
 +
is bounded in  $  D $
 +
and has asymptotic value  $  a $
 +
at  $  \zeta _ {0} \in D $,
 +
then it has at this point  $  a $
 +
as angular boundary value. Normality of a family  $  F = \{ f ( z) \} $
 +
of meromorphic functions  $  f ( z) $
 +
in a domain  $  G $
 +
can be characterized in terms of the so-called spherical derivative
 +
 
 +
$$
 +
\rho ( f ( z))  = \
 +
 
 +
\frac{| f ^ { \prime } ( z) | }{1 + | f ( z) |  ^ {2} }
 +
.
 +
$$
  
is open, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660201.png" /> has capacity zero, and every value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660202.png" /> is an asymptotic value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660203.png" /> either at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660204.png" /> or at every point of some sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660207.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660208.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660209.png" /> has capacity zero, then for every connected component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660210.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660211.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660212.png" /> the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660213.png" /> consists of at most two distinct values.
+
To be precise,  $  F $
 +
is a normal family if and only if the spherical derivatives  $  \rho ( f ( z)) $,  
 +
$  f \in F $,  
 +
are uniformly bounded inside  $  G $,  
 +
i.e. if for every compactum  $  K \subset  G $
 +
there is a constant  $  C = C ( K) $
 +
such that
  
Lindelöf's theorem has been proved using normal families (cf. [[Normal family|Normal family]]): If a holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660214.png" /> is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660215.png" /> and has asymptotic value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660216.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660217.png" />, then it has at this point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660218.png" /> as angular boundary value. Normality of a family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660219.png" /> of meromorphic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660220.png" /> in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660221.png" /> can be characterized in terms of the so-called spherical derivative
+
$$
 +
\rho ( f ( z))  \leq  \
 +
C ( K),\ \
 +
z \in K,\ \
 +
f \in F.
 +
$$
  
<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/c/c022/c022660/c022660222.png" /></td> </tr></table>
+
However, the most important occurrence of normal families in the theory of cluster sets is in the notion of a normal function. A function  $  f ( z) $,
 +
meromorphic in a simply-connected domain  $  G $,
 +
is called a normal function in  $  G $
 +
if the family  $  \{ f ( S ( z)) \} $,
 +
where  $  S $
 +
runs through the family of all conformal automorphisms of  $  G $,
 +
is normal; $  f ( z) $
 +
is normal in a multiply-connected domain  $  G $
 +
if it is normal on the universal covering surface of  $  G $.
 +
A function  $  f ( z) $,
 +
meromorphic in  $  D $,
 +
is normal if and only if there is a constant  $  C = C ( f  ) $,
 +
0 < C < \infty $,
 +
such that
  
To be precise, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660223.png" /> is a normal family if and only if the spherical derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660224.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660225.png" />, are uniformly bounded inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660226.png" />, i.e. if for every compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660227.png" /> there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660228.png" /> such that
+
$$
 +
\rho ( f ( z)) \
 +
| dz |  \leq  C
  
<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/c/c022/c022660/c022660229.png" /></td> </tr></table>
+
\frac{| dz | }{1 - | z |  ^ {2} }
 +
.
 +
$$
  
However, the most important occurrence of normal families in the theory of cluster sets is in the notion of a normal function. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660230.png" />, meromorphic in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660231.png" />, is called a normal function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660232.png" /> if the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660233.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660234.png" /> runs through the family of all conformal automorphisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660235.png" />, is normal; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660236.png" /> is normal in a multiply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660237.png" /> if it is normal on the universal covering surface of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660238.png" />. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660239.png" />, meromorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660240.png" />, is normal if and only if there is a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660241.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660242.png" />, such that
+
Here, the left-hand side is the line element in the so-called chordal metric on the Riemann sphere  $  \Omega $
 +
for the mapping  $  w = f ( z) $,
 +
while the expression  $  d \sigma ( z) = {| dz | } / {( 1 - | z |  ^ {2} ) } $
 +
is the hyperbolic metric of  $  D $.  
 +
Bounded holomorphic functions and meromorphic functions not taking three distinct values are normal, and certain properties of functions of the classes indicated carry over to arbitrary normal functions. E.g., the conclusion of Lindelöf's theorem holds for arbitrary normal functions. The class of all normal meromorphic functions in $  D $
 +
has some resemblance to the class of functions of bounded characteristic (cf. [[Function of bounded characteristic|Function of bounded characteristic]]). There are, however, essential differences. E.g., there exist normal meromorphic functions without asymptotic values, hence without radial boundary values, a fact which cannot hold for functions of bounded characteristic. G.R. MacLane [[#References|[7]]], [[#References|[9]]] conducted important studies on asymptotic values. MacLane's theory allows one to obtain new proofs of already known properties of normal functions. E.g., the set of points  $  \zeta \in \Gamma $
 +
at which a normal holomorphic function  $  f ( z) $
 +
has asymptotic values, hence angular boundary values, is dense on  $  \Gamma $.
  
<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/c/c022/c022660/c022660243.png" /></td> </tr></table>
+
The value distribution of meromorphic functions is closely connected with the notion of normality. A sequence  $  \{ z _ {n} \} $
 +
of points  $  z _ {n} $
 +
in  $  D $
 +
with  $  \lim\limits _ {n \rightarrow \infty }  | z _ {n} | = 1 $
 +
is called a  $  P $-
 +
sequence for a meromorphic function  $  f ( z) $
 +
in  $  D $
 +
if for every infinite subsequence  $  \{ z _ {n _ {k}  } \} $
 +
and every  $  \epsilon > 0 $
 +
the set
  
Here, the left-hand side is the line element in the so-called chordal metric on the Riemann sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660244.png" /> for the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660245.png" />, while the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660246.png" /> is the hyperbolic metric of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660247.png" />. Bounded holomorphic functions and meromorphic functions not taking three distinct values are normal, and certain properties of functions of the classes indicated carry over to arbitrary normal functions. E.g., the conclusion of Lindelöf's theorem holds for arbitrary normal functions. The class of all normal meromorphic functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660248.png" /> has some resemblance to the class of functions of bounded characteristic (cf. [[Function of bounded characteristic|Function of bounded characteristic]]). There are, however, essential differences. E.g., there exist normal meromorphic functions without asymptotic values, hence without radial boundary values, a fact which cannot hold for functions of bounded characteristic. G.R. MacLane [[#References|[7]]], [[#References|[9]]] conducted important studies on asymptotic values. MacLane's theory allows one to obtain new proofs of already known properties of normal functions. E.g., the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660249.png" /> at which a normal holomorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660250.png" /> has asymptotic values, hence angular boundary values, is dense on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660251.png" />.
+
$$
 +
\Omega \setminus  R
 +
\left (
 +
f, z; \
 +
\cup _ {k = 1 } ^  \infty 
 +
\{ {z \in D } : {
 +
\sigma ( z, z _ {n _ {k}  } )
 +
< \epsilon } \}
 +
\right )
 +
$$
  
The value distribution of meromorphic functions is closely connected with the notion of normality. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660252.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660253.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660254.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660255.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660257.png" />-sequence for a meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660258.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660259.png" /> if for every infinite subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660260.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660261.png" /> the set
+
contains at most two values. It has been proved that  $  f $
 +
has at least one  $  P $-
 +
sequence if and only if
  
<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/c/c022/c022660/c022660262.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {| z | \rightarrow 1 } \
 +
\sup \
 +
q _ {f} ( z)  = \
 +
+ \infty ,\ \
 +
q _ {f} ( z)  = \
 +
( 1 - | z |  ^ {2} )
 +
\rho ( f ( z)).
 +
$$
  
contains at most two values. It has been proved that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660263.png" /> has at least one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660264.png" />-sequence if and only if
+
Thus, the value distribution of the meromorphic function  $  f ( z) $
 +
is related to the structure of the cluster set of the continuous function  $  q _ {f} ( z) $.
  
<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/c/c022/c022660/c022660265.png" /></td> </tr></table>
+
Substantial progress has been made on the theory of cluster sets of general mappings  $  f:  D \rightarrow \Omega $,
 +
$  D = \{ {z } : {| z | < 1 } \} $.
 +
Already in 1955 the ambiguous point theorem was proved: Let  $  f: D \rightarrow \Omega $
 +
be an arbitrary mapping; then the points  $  \zeta \in \Gamma $
 +
at which one can draw two continuous curves  $  L _  \zeta  ^ {1} $
 +
and  $  L _  \zeta  ^ {2} $
 +
such that
  
Thus, the value distribution of the meromorphic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660266.png" /> is related to the structure of the cluster set of the continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660267.png" />.
+
$$
 +
C ( f, \zeta ; \
 +
L _  \zeta  ^ {1} )  \neq \
 +
C ( f, \zeta ; \
 +
L _  \zeta  ^ {2} ),
 +
$$
  
Substantial progress has been made on the theory of cluster sets of general mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660268.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660269.png" />. Already in 1955 the ambiguous point theorem was proved: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660270.png" /> be an arbitrary mapping; then the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660271.png" /> at which one can draw two continuous curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660272.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660273.png" /> such that
+
form a set that is at most countable. Collingwood's maximality theorem: Let $  L _ {0} $
 +
be an arbitrary continuum in  $  D $
 +
such that  $  L _ {0} \cap \Gamma = \{ z = 1 \} $,
 +
let  $  L _  \theta  $
 +
be the continuum obtained from  $  L _ {0} $
 +
by rotation over  $  \theta $
 +
around the coordinate origin and let  $  f: D \rightarrow \Omega $
 +
be an arbitrary mapping; then the points $  \zeta = e ^ {i \theta } \in \Gamma $
 +
at which
  
<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/c/c022/c022660/c022660274.png" /></td> </tr></table>
+
$$
 +
C ( f, \zeta ; \
 +
L _  \theta  )  \neq \
 +
C ( f, \zeta ; D),
 +
$$
  
form a set that is at most countable. Collingwood's maximality theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660275.png" /> be an arbitrary continuum in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660276.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660277.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660278.png" /> be the continuum obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660279.png" /> by rotation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660280.png" /> around the coordinate origin and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660281.png" /> be an arbitrary mapping; then the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660282.png" /> at which
+
form a set of the first category on  $  \Gamma $.  
 +
A point  $  \zeta \in \Gamma $
 +
is said to belong to the set  $  C ( f  ) $
 +
if the cluster set  $  C ( f, \zeta ; D) $
 +
coincides with the intersection
  
<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/c/c022/c022660/c022660283.png" /></td> </tr></table>
+
$$
 +
\cap C ( f, \zeta ; \
 +
\Delta ( \zeta ,\
 +
\phi _ {1} , \phi _ {2} ))
 +
$$
  
form a set of the first category on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660284.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660285.png" /> is said to belong to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660286.png" /> if the cluster set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660287.png" /> coincides with the intersection
+
over all angular domains with vertex  $  \zeta $.  
 +
It has been proved [[#References|[10]]] 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/c/c022/c022660/c022660288.png" /></td> </tr></table>
+
$$
 +
\Gamma  = \
 +
C ( f  ) \cup E
 +
$$
  
over all angular domains with vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660289.png" />. It has been proved [[#References|[10]]] that
+
for an arbitrary mapping  $  f: D \rightarrow \Omega $,
 +
where  $  E $
 +
is a set of the first category of type  $  F _  \sigma  $.  
 +
Conversely, for an arbitrary set  $  E \subset  \Gamma $
 +
of the first category and of type  $  F _  \sigma  $
 +
there exists a function  $  f $,
 +
holomorphic and bounded in  $  D $,
 +
for which  $  E = \Gamma \setminus  C ( f  ) $.  
 +
The set  $  C ( f  ) $
 +
is a subset of the set  $  K ( f  ) $
 +
of all  $  \zeta \in \Gamma $
 +
at which
  
<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/c/c022/c022660/c022660290.png" /></td> </tr></table>
+
$$
 +
C ( f, \zeta ; \Delta
 +
( \zeta , \phi _ {1} , \phi _ {2} ))  = \
 +
C ( f, \zeta ; \Delta
 +
( \zeta , \phi _ {1}  ^  \prime  , \phi _ {2}  ^  \prime  ))
 +
$$
  
for an arbitrary mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660291.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660292.png" /> is a set of the first category of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660293.png" />. Conversely, for an arbitrary set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660294.png" /> of the first category and of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660295.png" /> there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660296.png" />, holomorphic and bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660297.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660298.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660299.png" /> is a subset of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660300.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660301.png" /> at which
+
for any two angular domains  $  \Delta ( \zeta , \phi _ {1} , \phi _ {2} ) $
 +
and  $  \Delta ( \zeta , \phi _ {1}  ^  \prime  , \phi _ {2}  ^  \prime  ) $.  
 +
Let  $  E \subset  \Gamma $
 +
and  $  \zeta \in \Gamma $.  
 +
For a given  $  \epsilon > 0 $
 +
let  $  r ( \zeta , \epsilon , E) $
 +
denote the length of the largest open arc on  $  \Gamma $
 +
contained in the  $  \epsilon $-
 +
neighbourhood  $  \{ {e ^ {i \theta } } : {| \theta -  \mathop{\rm arg}  \zeta | < \epsilon } \} $
 +
of  $  \zeta $
 +
and not having points in common with  $  E $;
 +
if such an arc does not exist,  $  r ( \zeta , \epsilon , E) = 0 $.  
 +
A set $  E $
 +
is called porous on  $  \Gamma $
 +
if for any point  $  \zeta \in E $,
  
<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/c/c022/c022660/c022660302.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {\epsilon \rightarrow 0 } \
 +
\sup \
  
for any two angular domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660303.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660304.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660306.png" />. For a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660307.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660308.png" /> denote the length of the largest open arc on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660309.png" /> contained in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660310.png" />-neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660311.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660312.png" /> and not having points in common with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660313.png" />; if such an arc does not exist, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660314.png" />. A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660315.png" /> is called porous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660316.png" /> if for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660317.png" />,
+
\frac{r ( \zeta , \epsilon , E) } \epsilon
  
<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/c/c022/c022660/c022660318.png" /></td> </tr></table>
+
> 0;
 +
$$
  
a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660320.png" />-porous set is a union of at most countably many porous sets. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660321.png" />-porous set is of the first category and has linear measure zero. The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660322.png" /> is valid for any mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660323.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660324.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660325.png" />-porous set of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660326.png" />. Conversely, for an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660327.png" />-porous set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660328.png" /> there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660329.png" />, holomorphic and bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660330.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660331.png" />.
+
a $  \sigma $-
 +
porous set is a union of at most countably many porous sets. Every $  \sigma $-
 +
porous set is of the first category and has linear measure zero. The equality $  \Gamma = K ( f  ) \cup E $
 +
is valid for any mapping $  f: D \rightarrow \Omega $,  
 +
where $  E $
 +
is a $  \sigma $-
 +
porous set of type $  G _ {\delta \sigma }  $.  
 +
Conversely, for an arbitrary $  \sigma $-
 +
porous set $  E $
 +
there exists a function $  f $,  
 +
holomorphic and bounded in $  D $,  
 +
such that $  \Gamma \setminus  K ( f  ) \supset E $.
  
 
About the theory of cluster sets of functions of several complex variables see, e.g., [[#References|[15]]]–[[#References|[17]]].
 
About the theory of cluster sets of functions of several complex variables see, e.g., [[#References|[15]]]–[[#References|[17]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P. Painlevé,   "Leçons sur la théorie analytique des équations différentielles, professées à Stockholm (1895)" , Paris (1897)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B. Zoretti,   "Leçons sur la prolongement analytique" , Gauthier-Villars (1911)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.V. Golubev,   "Univalent analytic functions. Automorphic functions" , Moscow (1961) (In Russian)</TD></TR><TR><TD valign="top">[4]</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">[5]</TD> <TD valign="top"> K. Noshiro,   "Cluster sets" , Springer (1960)</TD></TR><TR><TD valign="top">[6]</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">[7]</TD> <TD valign="top"> G.R. MacLane,   "Asymptotic values of holomorphic functions" , ''Rice Univ. Studies, Math. Monographs'' , '''49''' : 1 , Rice Univ. , Houston (1963)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.I. Markushevich,   G.Ts. Tumarkin,   S.Ya. Khavinson, , ''Studies on comtemporary problems in the theory of functions of a complex variable'' , Moscow (1961) pp. 100–110 (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A. Lohwater,   "The boundary behaviour of analytic functions" ''Itogi Nauk. Mat. Anal.'' , '''10''' (1973) pp. 99–259 (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> E.P. Dolzhenko,   "Boundary properties of arbitrary functions" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''31''' : 1 (1967) pp. 3–14 (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> E.P. Dolzhenko,   "The metric properties of singular sets of holomorphic functions of several variables" ''Ann. of Math.'' , '''2''' (1976) pp. 191–201 (In Russian) (English summary)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> V.I. Gavrilov,   "Behavior of holomorphic functions along a chord in the unit disk" ''Soviet Math. Dokl.'' , '''15''' : 3 (1974) pp. 725–728 ''Dokl. Akad. Nauk SSSR'' , '''216''' : 1 (1974) pp. 21–23</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> A.N. Kanatnikov,   V.I. Gavrilov,   "Characterization of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660332.png" /> for meromorphic functions" ''Soviet Math. Dokl.'' , '''18''' : 2 (1977) pp. 270–272 ''Dokl. Akad. Nauk SSSR'' , '''233''' : 1 (1977) pp. 15–17</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A.N. Kanatnikov,   "A converse to Meier's theorem on meromorphic functions" ''Soviet Math. Dokl.'' , '''19''' : 1 (1978) pp. 162–165 ''Dokl. Akad. Nauk SSSR'' , '''238''' : 5 (1978) pp. 1043–1046</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> W. Rudin,   "Function theory in polydiscs" , Benjamin (1969)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> G.M. Khenkin,   E.M. Chirka,   "Boundary properties of holomorphic functions of several complex variables" ''J. Soviet Math.'' , '''5''' (1976) pp. 612–687 ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''4''' (1975) pp. 13–142</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> W. Rudin,   "Function theory in the unit ball in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022660/c022660333.png" />" , Springer (1980)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> P. Painlevé, "Leçons sur la théorie analytique des équations différentielles, professées à Stockholm (1895)" , Paris (1897)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> B. Zoretti, "Leçons sur la prolongement analytique" , Gauthier-Villars (1911)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V.V. Golubev, "Univalent analytic functions. Automorphic functions" , Moscow (1961) (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) {{MR|0083565}} {{ZBL|}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Noshiro, "Cluster sets" , Springer (1960) {{MR|0133464}} {{ZBL|0090.28801}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9 {{MR|0231999}} {{ZBL|0149.03003}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> G.R. MacLane, "Asymptotic values of holomorphic functions" , ''Rice Univ. Studies, Math. Monographs'' , '''49''' : 1 , Rice Univ. , Houston (1963)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> A.I. Markushevich, G.Ts. Tumarkin, S.Ya. Khavinson, , ''Studies on comtemporary problems in the theory of functions of a complex variable'' , Moscow (1961) pp. 100–110 (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> A. Lohwater, "The boundary behaviour of analytic functions" ''Itogi Nauk. Mat. Anal.'' , '''10''' (1973) pp. 99–259 (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> E.P. Dolzhenko, "Boundary properties of arbitrary functions" ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''31''' : 1 (1967) pp. 3–14 (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> E.P. Dolzhenko, "The metric properties of singular sets of holomorphic functions of several variables" ''Ann. of Math.'' , '''2''' (1976) pp. 191–201 (In Russian) (English summary)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> V.I. Gavrilov, "Behavior of holomorphic functions along a chord in the unit disk" ''Soviet Math. Dokl.'' , '''15''' : 3 (1974) pp. 725–728 ''Dokl. Akad. Nauk SSSR'' , '''216''' : 1 (1974) pp. 21–23</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> A.N. Kanatnikov, V.I. Gavrilov, "Characterization of the set $M(f)$ for meromorphic functions" ''Soviet Math. Dokl.'' , '''18''' : 2 (1977) pp. 270–272 ''Dokl. Akad. Nauk SSSR'' , '''233''' : 1 (1977) pp. 15–17 {{MR|437770}} {{ZBL|0377.30022}} {{ZBL|0373.30029}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A.N. Kanatnikov, "A converse to Meier's theorem on meromorphic functions" ''Soviet Math. Dokl.'' , '''19''' : 1 (1978) pp. 162–165 ''Dokl. Akad. Nauk SSSR'' , '''238''' : 5 (1978) pp. 1043–1046 {{MR|477055}} {{ZBL|}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> W. Rudin, "Function theory in polydiscs" , Benjamin (1969) {{MR|0255841}} {{ZBL|0177.34101}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> G.M. Khenkin, E.M. Chirka, "Boundary properties of holomorphic functions of several complex variables" ''J. Soviet Math.'' , '''5''' (1976) pp. 612–687 ''Itogi Nauk. i Tekhn. Sovrem. Probl. Mat.'' , '''4''' (1975) pp. 13–142</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> W. Rudin, "Function theory in the unit ball in $\CC^n$" , Springer (1980) {{MR|601594}} {{ZBL|0495.32001}} </TD></TR></table>
 
 
 
 
  
 
====Comments====
 
====Comments====
 
For the notions of linear Hausdorff measure and plane measure cf. [[Hausdorff measure|Hausdorff measure]]; for the chordal metric (also called spherical metric) cf. [[Extended complex plane|Extended complex plane]].
 
For the notions of linear Hausdorff measure and plane measure cf. [[Hausdorff measure|Hausdorff measure]]; for the chordal metric (also called spherical metric) cf. [[Extended complex plane|Extended complex plane]].

Latest revision as of 09:43, 26 March 2023


$ C ( f, z _ {0} ; S) $ of a function $ f : G \rightarrow \Omega $, defined on a domain $ G \subset \mathbf C $ with values in the Riemann sphere $ \Omega $, at a point $ z _ {0} \in \overline{G}\; $ with respect to a set $ S \subset G $, $ z _ {0} \in \overline{S}\; $

The set of values $ a \in \Omega $ for which there exists a sequence of points $ \{ z _ {n} \} _ {n = 1 } ^ \infty $, $ z _ {n} \in S $, $ \lim\limits _ {n \rightarrow \infty } z _ {n} = z _ {0} $, such that

$$ \lim\limits _ {n \rightarrow \infty } \ f ( z _ {n} ) = a. $$

Every number $ a \in C ( f, z _ {0} ; S) $ is called a cluster value of $ f $ at $ z _ {0} $ with respect to $ S $. The theory of cluster sets is a branch of function theory in which boundary properties of functions are studied in terms of topological and metric properties of various cluster sets.

If the entire domain $ G $ is taken for $ S $, one obtains the full cluster set $ C ( f, z _ {0} ; G) = C ( f, z _ {0} ) $; if the inclusion $ S \subset G $ is strict, the corresponding set $ C ( f, z _ {0} ; S) $ is sometimes called a partial cluster set. A full cluster set $ C ( f, z _ {0} ) $ is closed; if $ f $ is continuous on a set $ S $ that is locally connected at $ z _ {0} \in \overline{S}\; $, then the cluster set $ C ( f, z _ {0} ; S) $ is either degenerate, i.e. consists of a single point, or is a non-degenerate continuum. If $ C ( f, z _ {0} ; S) $ coincides with $ \Omega $, then it is called a total cluster set. A number $ a \in \Omega $ belongs to the set of recurrent values $ R ( f, z _ {0} ; S) $ of $ f $ at $ z _ {0} $ with respect to $ S $ if there is a sequence $ \{ z _ {n} \} $ of points $ z _ {n} \in S $, $ n = 1, 2 \dots $ $ \lim\limits _ {n \rightarrow \infty } z _ {n} = z _ {0} $, such that $ a = f ( z _ {n} ) $, $ n = 1, 2 , . . . $. One always has $ R ( f, z _ {0} ; S) \subset C ( f, z _ {0} ; S) $. If for some $ a \in \Omega $ there is a path $ L $: $ z = z ( t) $, $ 0 \leq t < 1 $, in $ G $ ending at a point $ z _ {0} $, $ z _ {0} \in \overline{G}\; $, $ \lim\limits _ {t \rightarrow 1 } z ( t) = z _ {0} $, and such that $ \lim\limits _ {t \rightarrow 1 } f ( z ( t)) = a $, then $ a $ is called an asymptotic value of $ f $ at $ z _ {0} $( along $ L $). The asymptotic set $ A ( f, z _ {0} ; G) $ is the set of all asymptotic values of $ f $ at $ z _ {0} $.

The notion of a cluster set was clearly formulated for the first time by P. Painlevé in 1895 (he called it the "region of indeterminacy" , cf. [1]) in connection with studying an analytic function near one of its singular points and with classifying singularities of such functions. At that time one basically studied three, geometrically most simple, cases in the theory of cluster sets: a) $ z _ {0} $ is an isolated point of the boundary $ \partial G $ or an interior point of $ G $; b) $ G = D = \{ {z } : {| z | < 1 } \} $ is the unit disc or, in general, a Jordan domain, and $ z _ {0} $ is a point on the boundary $ \Gamma = \partial D $; and c) the boundary $ E = \partial G $ is an everywhere-discontinuous compactum in the plane (i.e. a totally-disconnected compact set) and $ z _ {0} \in E $. A number of classical results in complex function theory have a formulation in terms of cluster sets. E.g., the Sokhotskii theorem, in a somewhat stronger form, states: If $ z _ {0} $ is an isolated point of an everywhere-discontinuous compactum $ E \subset G $ and $ f $ is a meromorphic function on $ G \setminus E $, then the cluster set $ C ( f, z _ {0} ; G \setminus E) $ is either degenerate or total. The Picard theorem, supplementing it, states that if $ C ( f, z _ {0} ; G \setminus E) $ is total, i.e. if $ z _ {0} $ is an essential singular point, then the set $ CR ( f, z _ {0} ; G \setminus E) = \Omega \setminus R ( f, z _ {0} ; G \setminus E) $ contains at most two different values. Also, in this case

$$ CR ( f, z _ {0} ; G \setminus E) \subset \ A ( f, z _ {0} ; G \setminus E) $$

(the Iversen theorem).

The main result related to the theory of the behaviour of meromorphic functions near "thin" boundaries (the Painlevé theory) is (cf. [1], [2]): If a set $ E \subset G $ has linear Hausdorff measure zero, $ \mu ( E) = \mu _ {1} ( E) = 0 $, and the function $ f $ is meromorphic in $ G \setminus E $, then for every point $ z _ {0} \in E $ the cluster set $ C ( f, z _ {0} ; G \setminus E) $ is either degenerate or total; moreover, in the first case $ f $ is also meromorphic at $ z _ {0} $. Thus, a point $ z _ {0} \in E $ for which the cluster set $ C ( f, z _ {0} ; G \setminus E) $ is degenerate is a removable singular point of $ f $; the study of removable sets of various function classes can be regarded as a branch of the theory of cluster sets.

Golubev's theorem is an important strengthening of the theorem of Picard: If $ E \subset G $, $ \mu ( E) = 0 $ and $ f $ is meromorphic in $ G \setminus E $, then the set $ CR ( f, z _ {0} ; G \setminus E) $ has analytic capacity zero at every essential singular point $ z _ {0} \in E $( hence its plane measure $ \mu _ {2} ( CR) = 0 $).

The work of P. Fatou (1906) on boundary values of functions $ f ( z) $ holomorphic in the unit disc $ D = \{ {z } : {| z | < 1 } \} $ was the starting point for the theory of cluster sets in the case of continuous boundaries. If such a function $ f $ is bounded in $ D $, then almost-everywhere (in the sense of the Lebesgue measure) on the circle $ \Gamma = \{ {z } : {| z | = 1 } \} $ it has radial and angular (non-tangential) boundary values (Fatou's theorem). Let $ \zeta = e ^ {i \theta } \in \Gamma $ be an arbitrary point; denote by $ h ( \zeta , \phi ) $ the chord of $ D $ ending at $ \zeta $ and forming with the radius at $ \zeta $ an angle $ \phi $, $ - \pi /2 < \phi < \pi /2 $. Let $ \Delta ( \zeta , \phi _ {1} , \phi _ {2} ) $ be the angular domain with vertex $ \zeta \in \Gamma $, consisting of those points of $ D $ lying between the chords

$$ h ( \zeta , \phi _ {1} ) \ \ \textrm{ and } \ \ h ( \zeta , \phi _ {2} ),\ \ {- \frac \pi {2} } < \phi _ {1} < \phi _ {2} < { \frac \pi {2} } . $$

A point $ \zeta \in \Gamma $ is called a Fatou point, and belongs to the set $ F ( f ) $, if the union

$$ \cup C ( f, \zeta ; \Delta ( \zeta , \phi _ {1} , \phi _ {2} )) $$

over all angular domains $ \Delta ( \zeta , \phi _ {1} , \phi _ {2} ) $ consists of a single value $ f ( e ^ {i \theta } ) $, which is called the angular boundary value of $ f $ at $ \zeta $. Another formulation of Fatou's theorem: For a bounded holomorphic function $ f $ in $ D $ the decomposition $ \Gamma = F ( f ) \cup E $, $ \mathop{\rm mes} E = 0 $, holds. This result is supplemented by the F. and M. Riesz uniqueness theorem (1916): If $ f $ is holomorphic and bounded in $ D $ and if on some set $ M \subset F ( f ) $, $ \mathop{\rm mes} M > 0 $, it has angular boundary values $ f ( \zeta ) = a $, $ \zeta \in M $, then $ f ( z) \equiv a $. This statement was proved, independently, by N.N. Luzin and I.I. Privalov (1919), who obtained an essential generalization of it to the case of arbitrary meromorphic functions. In the same year they published a boundary uniqueness theorem for the case of radial boundary values: If a function $ f $, holomorphic in $ D $, has the same radial boundary value $ a \in \Omega $ on a set $ M $ of the second category and metrically dense on some arc $ \gamma \subset \Gamma $, i.e. if $ \lim\limits _ {r \rightarrow 1 } f ( re ^ {i \theta } ) = a $, $ e ^ {i \theta } \in M $, then $ f ( z) \equiv a $.

Privalov, in 1936, noted that the statement $ f ( z) \equiv \textrm{ const } $ remains true also when the values $ a _ \zeta = \lim\limits _ {r \rightarrow 1 } f ( re ^ {i \theta } ) $ are not necessarily equal at the points $ \zeta = e ^ {i \theta } \in M $, but belong to a set of (logarithmic) capacity zero. The basic idea and the elements of the proof of the Luzin–Privalov theorem are applicable in the general case of continuous mappings $ f $ of $ D $, which was subsequently used in many papers.

A point $ \zeta \in \Gamma = \{ {z } : {| z | = 1 } \} $ is called a Plessner point, and belongs to the set $ I ( f ) $, if the intersection

$$ \cap C ( f, \zeta ; \Delta ( \zeta , \phi _ {1} , \phi _ {2} )) $$

over all angular domains $ \Delta ( \zeta , \phi _ {1} , \phi _ {2} ) $ with vertex $ \zeta $ coincides with $ \Omega $. A.I. Plessner proved (1927) that for a meromorphic function $ f $ in $ D $ almost-all points of the boundary $ \Gamma $ belong either to $ F ( f ) $ or to $ I ( f ) $, i.e. $ \Gamma = F ( f ) \cup I ( f ) \cup E $, $ \mathop{\rm mes} E = 0 $. A point $ \zeta \in \Gamma $ is called a Meier point, and belongs to $ M ( f ) $, if $ C ( f, \zeta ; D) \neq \Omega $ and if the intersection of the chordal cluster sets, $ \cap C ( f, \zeta ; h ( \zeta , \phi )) $, over all chords drawn at $ \zeta $, coincides with $ C ( f, \zeta ; D) $. K. Meier established (1961) the following analogue of Plessner's theorem in terms of Baire categories: If $ f $ is meromorphic in $ D $, then all points of the boundary $ \Gamma $, with the possible exception of a set $ E $ of the first category, belong to the union $ M ( f ) \cup I ( f ) $. A more precise statement of Meier's theorem has been obtained, in which $ E $ is a set of the first category and of type $ F _ \sigma $( cf. [12][14], in which generalizations of Plessner's and Meier's theorems have been obtained, and in which a converse of Meier's theorem and a characterization of $ M ( f ) $ have been given).

The work of Fatou served as an original source for the development of fundamental research on boundary properties of analytic functions. The studies of F. and M. Riesz, Luzin, Privalov, R. Nevanlinna, Plessner, V.I. Smirnov, and others were conducted independently of the ideas of Painlevé, and the use of methods related to measure and integration theory, including the notion of Baire categories, is characteristic for them (cf. [4][9]).

The basic objects of study for F. Iversen and W. Gross were meromorphic functions $ f $ in domains $ D $ with a Jordan boundary $ \Gamma = \partial D $. At an arbitrary point $ \zeta _ {0} \in \Gamma $, the boundary cluster set $ C ( f, \zeta _ {0} ; \Gamma ) $ is defined as follows: If $ M _ {r} $ denotes the closure of the union $ \cup C ( f, \zeta ; D) $ over all points

$$ \zeta \in \ ( \Gamma \setminus \{ \zeta _ {0} \} ) \cap \{ {z } : { | z - \zeta _ {0} | < r } \} , $$

then $ C ( f, \zeta _ {0} ; \Gamma ) = \cap _ {r > 0 } M _ {r} $. One of the main theorems obtained, independently, by them asserts that, under the conditions stated, the set

$$ C _ {i} ( f,\ \zeta _ {0} ; D) = \ C ( f, \zeta _ {0} ; D) \setminus C ( f, \zeta _ {0} ; \Gamma ) $$

is open (for any $ \zeta _ {0} \in \Gamma $), and all values $ a \in C _ {i} ( f, \zeta _ {0} ; D) $, with possibly two exceptions, belong to the set of recurrent values $ R ( f, \zeta _ {0} ; D) $. Moreover, every exceptional value (if existing) is an asymptotic value of $ f $ at $ \zeta _ {0} $.

The research of Iversen and Gross obtained a further development in the work of A. Beurling, W. Seidel (who in 1932 also introduced the term "cluster set" ) and others (cf. [5][9]). They basically considered the case when $ \zeta _ {0} $ belongs to a "small" set $ E $ on the boundary $ \Gamma $, having zero linear measure or zero capacity, and studied the cluster set $ C ( f, \zeta _ {0} ; \Gamma \setminus E) $, defined analogously to $ C ( f, \zeta _ {0} ; \Gamma ) $. Methods of potential theory are also used in these studies.

The most recent results in this direction are stated below for the case of the disc $ D = \{ {z } : {| z | < 1 } \} $. Suppose a set $ E $ on an arc $ \gamma $ of the boundary $ \Gamma $ of $ D $ having $ \mathop{\rm mes} E = 0 $ is fixed, and let $ \zeta _ {0} \in E $. To every point $ \zeta \in \gamma \setminus E $ one assigns a Jordan arc $ \Lambda _ \zeta \subset D $ ending at $ \zeta $. Let $ M _ {r} ^ {*} $ be the closure of the union $ \cup C ( f, \zeta ; \Lambda _ \zeta ) $ over all points

$$ \zeta \in \ ( \gamma \setminus E) \cap \{ {z } : { | z - \zeta _ {0} | < r } \} $$

and suppose

$$ C ^ {*} ( f, \zeta _ {0} ; \ \Gamma \setminus E) = \ \cap _ {r > 0 } M _ {r} ^ {*} . $$

Then the set

$$ S ( \zeta _ {0} ) = \ C ( f, \zeta _ {0} ; D) \setminus C ^ {*} ( f, \zeta _ {0} ; \ \Gamma \setminus E) $$

is open, the set $ S ( \zeta _ {0} ) \setminus R ( f, \zeta _ {0} ; D) $ has capacity zero, and every value $ a \in S ( \zeta _ {0} ) \setminus R ( f, \zeta _ {0} ; D) $ is an asymptotic value of $ f $ either at $ \zeta _ {0} $ or at every point of some sequence $ \{ \zeta _ {n} \} $, $ \zeta _ {n} \in \Gamma $, $ n = 1, 2 \dots $ $ \lim\limits _ {n \rightarrow \infty } \zeta _ {n} = \zeta _ {0} $. If $ E $ has capacity zero, then for every connected component $ S _ {k} ( \zeta _ {0} ) $, $ k = 1, 2 \dots $ of $ S ( \zeta _ {0} ) $ the set $ S _ {k} ( \zeta _ {0} ) \setminus R ( f, \zeta _ {0} ; D) $ consists of at most two distinct values.

Lindelöf's theorem has been proved using normal families (cf. Normal family): If a holomorphic function $ f $ is bounded in $ D $ and has asymptotic value $ a $ at $ \zeta _ {0} \in D $, then it has at this point $ a $ as angular boundary value. Normality of a family $ F = \{ f ( z) \} $ of meromorphic functions $ f ( z) $ in a domain $ G $ can be characterized in terms of the so-called spherical derivative

$$ \rho ( f ( z)) = \ \frac{| f ^ { \prime } ( z) | }{1 + | f ( z) | ^ {2} } . $$

To be precise, $ F $ is a normal family if and only if the spherical derivatives $ \rho ( f ( z)) $, $ f \in F $, are uniformly bounded inside $ G $, i.e. if for every compactum $ K \subset G $ there is a constant $ C = C ( K) $ such that

$$ \rho ( f ( z)) \leq \ C ( K),\ \ z \in K,\ \ f \in F. $$

However, the most important occurrence of normal families in the theory of cluster sets is in the notion of a normal function. A function $ f ( z) $, meromorphic in a simply-connected domain $ G $, is called a normal function in $ G $ if the family $ \{ f ( S ( z)) \} $, where $ S $ runs through the family of all conformal automorphisms of $ G $, is normal; $ f ( z) $ is normal in a multiply-connected domain $ G $ if it is normal on the universal covering surface of $ G $. A function $ f ( z) $, meromorphic in $ D $, is normal if and only if there is a constant $ C = C ( f ) $, $ 0 < C < \infty $, such that

$$ \rho ( f ( z)) \ | dz | \leq C \frac{| dz | }{1 - | z | ^ {2} } . $$

Here, the left-hand side is the line element in the so-called chordal metric on the Riemann sphere $ \Omega $ for the mapping $ w = f ( z) $, while the expression $ d \sigma ( z) = {| dz | } / {( 1 - | z | ^ {2} ) } $ is the hyperbolic metric of $ D $. Bounded holomorphic functions and meromorphic functions not taking three distinct values are normal, and certain properties of functions of the classes indicated carry over to arbitrary normal functions. E.g., the conclusion of Lindelöf's theorem holds for arbitrary normal functions. The class of all normal meromorphic functions in $ D $ has some resemblance to the class of functions of bounded characteristic (cf. Function of bounded characteristic). There are, however, essential differences. E.g., there exist normal meromorphic functions without asymptotic values, hence without radial boundary values, a fact which cannot hold for functions of bounded characteristic. G.R. MacLane [7], [9] conducted important studies on asymptotic values. MacLane's theory allows one to obtain new proofs of already known properties of normal functions. E.g., the set of points $ \zeta \in \Gamma $ at which a normal holomorphic function $ f ( z) $ has asymptotic values, hence angular boundary values, is dense on $ \Gamma $.

The value distribution of meromorphic functions is closely connected with the notion of normality. A sequence $ \{ z _ {n} \} $ of points $ z _ {n} $ in $ D $ with $ \lim\limits _ {n \rightarrow \infty } | z _ {n} | = 1 $ is called a $ P $- sequence for a meromorphic function $ f ( z) $ in $ D $ if for every infinite subsequence $ \{ z _ {n _ {k} } \} $ and every $ \epsilon > 0 $ the set

$$ \Omega \setminus R \left ( f, z; \ \cup _ {k = 1 } ^ \infty \{ {z \in D } : { \sigma ( z, z _ {n _ {k} } ) < \epsilon } \} \right ) $$

contains at most two values. It has been proved that $ f $ has at least one $ P $- sequence if and only if

$$ \lim\limits _ {| z | \rightarrow 1 } \ \sup \ q _ {f} ( z) = \ + \infty ,\ \ q _ {f} ( z) = \ ( 1 - | z | ^ {2} ) \rho ( f ( z)). $$

Thus, the value distribution of the meromorphic function $ f ( z) $ is related to the structure of the cluster set of the continuous function $ q _ {f} ( z) $.

Substantial progress has been made on the theory of cluster sets of general mappings $ f: D \rightarrow \Omega $, $ D = \{ {z } : {| z | < 1 } \} $. Already in 1955 the ambiguous point theorem was proved: Let $ f: D \rightarrow \Omega $ be an arbitrary mapping; then the points $ \zeta \in \Gamma $ at which one can draw two continuous curves $ L _ \zeta ^ {1} $ and $ L _ \zeta ^ {2} $ such that

$$ C ( f, \zeta ; \ L _ \zeta ^ {1} ) \neq \ C ( f, \zeta ; \ L _ \zeta ^ {2} ), $$

form a set that is at most countable. Collingwood's maximality theorem: Let $ L _ {0} $ be an arbitrary continuum in $ D $ such that $ L _ {0} \cap \Gamma = \{ z = 1 \} $, let $ L _ \theta $ be the continuum obtained from $ L _ {0} $ by rotation over $ \theta $ around the coordinate origin and let $ f: D \rightarrow \Omega $ be an arbitrary mapping; then the points $ \zeta = e ^ {i \theta } \in \Gamma $ at which

$$ C ( f, \zeta ; \ L _ \theta ) \neq \ C ( f, \zeta ; D), $$

form a set of the first category on $ \Gamma $. A point $ \zeta \in \Gamma $ is said to belong to the set $ C ( f ) $ if the cluster set $ C ( f, \zeta ; D) $ coincides with the intersection

$$ \cap C ( f, \zeta ; \ \Delta ( \zeta ,\ \phi _ {1} , \phi _ {2} )) $$

over all angular domains with vertex $ \zeta $. It has been proved [10] that

$$ \Gamma = \ C ( f ) \cup E $$

for an arbitrary mapping $ f: D \rightarrow \Omega $, where $ E $ is a set of the first category of type $ F _ \sigma $. Conversely, for an arbitrary set $ E \subset \Gamma $ of the first category and of type $ F _ \sigma $ there exists a function $ f $, holomorphic and bounded in $ D $, for which $ E = \Gamma \setminus C ( f ) $. The set $ C ( f ) $ is a subset of the set $ K ( f ) $ of all $ \zeta \in \Gamma $ at which

$$ C ( f, \zeta ; \Delta ( \zeta , \phi _ {1} , \phi _ {2} )) = \ C ( f, \zeta ; \Delta ( \zeta , \phi _ {1} ^ \prime , \phi _ {2} ^ \prime )) $$

for any two angular domains $ \Delta ( \zeta , \phi _ {1} , \phi _ {2} ) $ and $ \Delta ( \zeta , \phi _ {1} ^ \prime , \phi _ {2} ^ \prime ) $. Let $ E \subset \Gamma $ and $ \zeta \in \Gamma $. For a given $ \epsilon > 0 $ let $ r ( \zeta , \epsilon , E) $ denote the length of the largest open arc on $ \Gamma $ contained in the $ \epsilon $- neighbourhood $ \{ {e ^ {i \theta } } : {| \theta - \mathop{\rm arg} \zeta | < \epsilon } \} $ of $ \zeta $ and not having points in common with $ E $; if such an arc does not exist, $ r ( \zeta , \epsilon , E) = 0 $. A set $ E $ is called porous on $ \Gamma $ if for any point $ \zeta \in E $,

$$ \lim\limits _ {\epsilon \rightarrow 0 } \ \sup \ \frac{r ( \zeta , \epsilon , E) } \epsilon > 0; $$

a $ \sigma $- porous set is a union of at most countably many porous sets. Every $ \sigma $- porous set is of the first category and has linear measure zero. The equality $ \Gamma = K ( f ) \cup E $ is valid for any mapping $ f: D \rightarrow \Omega $, where $ E $ is a $ \sigma $- porous set of type $ G _ {\delta \sigma } $. Conversely, for an arbitrary $ \sigma $- porous set $ E $ there exists a function $ f $, holomorphic and bounded in $ D $, such that $ \Gamma \setminus K ( f ) \supset E $.

About the theory of cluster sets of functions of several complex variables see, e.g., [15][17].

References

[1] P. Painlevé, "Leçons sur la théorie analytique des équations différentielles, professées à Stockholm (1895)" , Paris (1897)
[2] B. Zoretti, "Leçons sur la prolongement analytique" , Gauthier-Villars (1911)
[3] V.V. Golubev, "Univalent analytic functions. Automorphic functions" , Moscow (1961) (In Russian)
[4] I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) MR0083565
[5] K. Noshiro, "Cluster sets" , Springer (1960) MR0133464 Zbl 0090.28801
[6] E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9 MR0231999 Zbl 0149.03003
[7] G.R. MacLane, "Asymptotic values of holomorphic functions" , Rice Univ. Studies, Math. Monographs , 49 : 1 , Rice Univ. , Houston (1963)
[8] A.I. Markushevich, G.Ts. Tumarkin, S.Ya. Khavinson, , Studies on comtemporary problems in the theory of functions of a complex variable , Moscow (1961) pp. 100–110 (In Russian)
[9] A. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian)
[10] E.P. Dolzhenko, "Boundary properties of arbitrary functions" Izv. Akad. Nauk SSSR Ser. Mat. , 31 : 1 (1967) pp. 3–14 (In Russian)
[11] E.P. Dolzhenko, "The metric properties of singular sets of holomorphic functions of several variables" Ann. of Math. , 2 (1976) pp. 191–201 (In Russian) (English summary)
[12] V.I. Gavrilov, "Behavior of holomorphic functions along a chord in the unit disk" Soviet Math. Dokl. , 15 : 3 (1974) pp. 725–728 Dokl. Akad. Nauk SSSR , 216 : 1 (1974) pp. 21–23
[13] A.N. Kanatnikov, V.I. Gavrilov, "Characterization of the set $M(f)$ for meromorphic functions" Soviet Math. Dokl. , 18 : 2 (1977) pp. 270–272 Dokl. Akad. Nauk SSSR , 233 : 1 (1977) pp. 15–17 MR437770 Zbl 0377.30022 Zbl 0373.30029
[14] A.N. Kanatnikov, "A converse to Meier's theorem on meromorphic functions" Soviet Math. Dokl. , 19 : 1 (1978) pp. 162–165 Dokl. Akad. Nauk SSSR , 238 : 5 (1978) pp. 1043–1046 MR477055
[15] W. Rudin, "Function theory in polydiscs" , Benjamin (1969) MR0255841 Zbl 0177.34101
[16] G.M. Khenkin, E.M. Chirka, "Boundary properties of holomorphic functions of several complex variables" J. Soviet Math. , 5 (1976) pp. 612–687 Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 4 (1975) pp. 13–142
[17] W. Rudin, "Function theory in the unit ball in $\CC^n$" , Springer (1980) MR601594 Zbl 0495.32001

Comments

For the notions of linear Hausdorff measure and plane measure cf. Hausdorff measure; for the chordal metric (also called spherical metric) cf. Extended complex plane.

How to Cite This Entry:
Cluster set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cluster_set&oldid=16145
This article was adapted from an original article by V.I. GavrilovE.D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article