Cluster set
of a function
, defined on a domain
with values in the Riemann sphere
, at a point
with respect to a set
,
The set of values for which there exists a sequence of points
,
,
, such that
![]() |
Every number is called a cluster value of
at
with respect to
. 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 is taken for
, one obtains the full cluster set
; if the inclusion
is strict, the corresponding set
is sometimes called a partial cluster set. A full cluster set
is closed; if
is continuous on a set
that is locally connected at
, then the cluster set
is either degenerate, i.e. consists of a single point, or is a non-degenerate continuum. If
coincides with
, then it is called a total cluster set. A number
belongs to the set of recurrent values
of
at
with respect to
if there is a sequence
of points
,
, such that
,
. One always has
. If for some
there is a path
:
,
, in
ending at a point
,
,
, and such that
, then
is called an asymptotic value of
at
(along
). The asymptotic set
is the set of all asymptotic values of
at
.
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) is an isolated point of the boundary
or an interior point of
; b)
is the unit disc or, in general, a Jordan domain, and
is a point on the boundary
; and c) the boundary
is an everywhere-discontinuous compactum in the plane (i.e. a totally-disconnected compact set) and
. 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
is an isolated point of an everywhere-discontinuous compactum
and
is a meromorphic function on
, then the cluster set
is either degenerate or total. The Picard theorem, supplementing it, states that if
is total, i.e. if
is an essential singular point, then the set
contains at most two different values. Also, in this case
![]() |
(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 has linear Hausdorff measure zero,
, and the function
is meromorphic in
, then for every point
the cluster set
is either degenerate or total; moreover, in the first case
is also meromorphic at
. Thus, a point
for which the cluster set
is degenerate is a removable singular point of
; 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 ,
and
is meromorphic in
, then the set
has analytic capacity zero at every essential singular point
(hence its plane measure
).
The work of P. Fatou (1906) on boundary values of functions holomorphic in the unit disc
was the starting point for the theory of cluster sets in the case of continuous boundaries. If such a function
is bounded in
, then almost-everywhere (in the sense of the Lebesgue measure) on the circle
it has radial and angular (non-tangential) boundary values (Fatou's theorem). Let
be an arbitrary point; denote by
the chord of
ending at
and forming with the radius at
an angle
,
. Let
be the angular domain with vertex
, consisting of those points of
lying between the chords
![]() |
A point is called a Fatou point, and belongs to the set
, if the union
![]() |
over all angular domains consists of a single value
, which is called the angular boundary value of
at
. Another formulation of Fatou's theorem: For a bounded holomorphic function
in
the decomposition
,
, holds. This result is supplemented by the F. and M. Riesz uniqueness theorem (1916): If
is holomorphic and bounded in
and if on some set
,
, it has angular boundary values
,
, then
. 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
, holomorphic in
, has the same radial boundary value
on a set
of the second category and metrically dense on some arc
, i.e. if
,
, then
.
Privalov, in 1936, noted that the statement remains true also when the values
are not necessarily equal at the points
, 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
of
, which was subsequently used in many papers.
A point is called a Plessner point, and belongs to the set
, if the intersection
![]() |
over all angular domains with vertex
coincides with
. A.I. Plessner proved (1927) that for a meromorphic function
in
almost-all points of the boundary
belong either to
or to
, i.e.
,
. A point
is called a Meier point, and belongs to
, if
and if the intersection of the chordal cluster sets,
, over all chords drawn at
, coincides with
. K. Meier established (1961) the following analogue of Plessner's theorem in terms of Baire categories: If
is meromorphic in
, then all points of the boundary
, with the possible exception of a set
of the first category, belong to the union
. A more precise statement of Meier's theorem has been obtained, in which
is a set of the first category and of type
(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
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 in domains
with a Jordan boundary
. At an arbitrary point
, the boundary cluster set
is defined as follows: If
denotes the closure of the union
over all points
![]() |
then . One of the main theorems obtained, independently, by them asserts that, under the conditions stated, the set
![]() |
is open (for any ), and all values
, with possibly two exceptions, belong to the set of recurrent values
. Moreover, every exceptional value (if existing) is an asymptotic value of
at
.
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 belongs to a "small" set
on the boundary
, having zero linear measure or zero capacity, and studied the cluster set
, defined analogously to
. 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 . Suppose a set
on an arc
of the boundary
of
having
is fixed, and let
. To every point
one assigns a Jordan arc
ending at
. Let
be the closure of the union
over all points
![]() |
and suppose
![]() |
Then the set
![]() |
is open, the set has capacity zero, and every value
is an asymptotic value of
either at
or at every point of some sequence
,
,
. If
has capacity zero, then for every connected component
,
of
the set
consists of at most two distinct values.
Lindelöf's theorem has been proved using normal families (cf. Normal family): If a holomorphic function is bounded in
and has asymptotic value
at
, then it has at this point
as angular boundary value. Normality of a family
of meromorphic functions
in a domain
can be characterized in terms of the so-called spherical derivative
![]() |
To be precise, is a normal family if and only if the spherical derivatives
,
, are uniformly bounded inside
, i.e. if for every compactum
there is a constant
such that
![]() |
However, the most important occurrence of normal families in the theory of cluster sets is in the notion of a normal function. A function , meromorphic in a simply-connected domain
, is called a normal function in
if the family
, where
runs through the family of all conformal automorphisms of
, is normal;
is normal in a multiply-connected domain
if it is normal on the universal covering surface of
. A function
, meromorphic in
, is normal if and only if there is a constant
,
, such that
![]() |
Here, the left-hand side is the line element in the so-called chordal metric on the Riemann sphere for the mapping
, while the expression
is the hyperbolic metric of
. 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
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
at which a normal holomorphic function
has asymptotic values, hence angular boundary values, is dense on
.
The value distribution of meromorphic functions is closely connected with the notion of normality. A sequence of points
in
with
is called a
-sequence for a meromorphic function
in
if for every infinite subsequence
and every
the set
![]() |
contains at most two values. It has been proved that has at least one
-sequence if and only if
![]() |
Thus, the value distribution of the meromorphic function is related to the structure of the cluster set of the continuous function
.
Substantial progress has been made on the theory of cluster sets of general mappings ,
. Already in 1955 the ambiguous point theorem was proved: Let
be an arbitrary mapping; then the points
at which one can draw two continuous curves
and
such that
![]() |
form a set that is at most countable. Collingwood's maximality theorem: Let be an arbitrary continuum in
such that
, let
be the continuum obtained from
by rotation over
around the coordinate origin and let
be an arbitrary mapping; then the points
at which
![]() |
form a set of the first category on . A point
is said to belong to the set
if the cluster set
coincides with the intersection
![]() |
over all angular domains with vertex . It has been proved [10] that
![]() |
for an arbitrary mapping , where
is a set of the first category of type
. Conversely, for an arbitrary set
of the first category and of type
there exists a function
, holomorphic and bounded in
, for which
. The set
is a subset of the set
of all
at which
![]() |
for any two angular domains and
. Let
and
. For a given
let
denote the length of the largest open arc on
contained in the
-neighbourhood
of
and not having points in common with
; if such an arc does not exist,
. A set
is called porous on
if for any point
,
![]() |
a -porous set is a union of at most countably many porous sets. Every
-porous set is of the first category and has linear measure zero. The equality
is valid for any mapping
, where
is a
-porous set of type
. Conversely, for an arbitrary
-porous set
there exists a function
, holomorphic and bounded in
, such that
.
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) |
[5] | K. Noshiro, "Cluster sets" , Springer (1960) |
[6] | E.F. Collingwood, A.J. Lohwater, "The theory of cluster sets" , Cambridge Univ. Press (1966) pp. Chapt. 9 |
[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 ![]() |
[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 |
[15] | W. Rudin, "Function theory in polydiscs" , Benjamin (1969) |
[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 ![]() |
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.
Cluster set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cluster_set&oldid=16145