ETNA
Electronic Transactions on Numerical Analysis.
Volume 25, pp. 27-40, 2006.
Copyright  2006, Kent State University.
ISSN 1068-9613.
Kent State University
etna@mcs.kent.edu
ON THE SUPPORT OF THE EQUILIBRIUM MEASURE FOR ARCS OF THE
UNIT CIRCLE AND FOR REAL INTERVALS
D. BENKO , S. B. DAMELIN , AND P. D. DRAGNEV
Dedicated to Ed Saff on the occasion of his 60th birthday
Abstract. We study the support of the equilibrium measure for weights defined on arcs of the unit circle and on
intervals of the compactified real line. We provide several conditions to ensure that the support of the equilibrium
measure is one interval or one arc.
Key words. logarthmic potential theory, external fields, equilibrium measure, equilibrium support
AMS subject classifications. 31A15, 30C15, 78A30
1. Introduction. In recent years, equilibrium measures with external fields have found
an increasing number of applications in a variety of areas ranging from diverse subjects such
as orthogonal polynomials, weighted Fekete points, numerical conformal mappings, weighted
polynomial approximation, rational and Pade approximation, integrable systems, random matrix theory and random permutations. We refer the reader to the references [1, 2, 4, 7, 8, 12,
13, 14, 16, 17, 19, 20, 21] and those listed therein for a comprehensive account of these
numerous, vast and interesting applications.
1.1. Potential-theoretic preliminaries and definitions. With a compact set
and
lower semi-continuous external field
, we set
and call a
weight associated with , provided the set
!"#$&%
(')
+*-,/.0
123,4%6587!9
has positive logarithmic capacity. With an external field (or a weight ), we associate the
weighted energy of a Borel probability measure : on as
J % :N=K#%O
;< =:"%> ?!@2?A@CBEDGF I J I H J
N
:
LK 2 %2=K#%4M M
The equilibrium
in the presence of an external field , is the unique probability
< on measure
measure :
minimizing the weighted energy among all probability measures on .
Thus,
; < P: < %>8Q/REST* ; < P:"%6
&:U.WVXYZ%94
where VX[Z% denotes the class
VX[Z%N\*]:U
&: is a Borel probability measure on ^94O
For more details on these topics we refer the reader to the seminal monograph of E. B. Saff
and V. Totik [17].
_
Received May 14, 2005. Accepted for publication January 6, 2006. Recommended by D. Lubinsky.
Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101
(dbenko2006@yahoo.com).
Institute for Mathematics and its Applications, University of Minnesota, 400 Lind Hall, 207 Church Hill, S.E.,
Minneapolis, MN 55455 (damelin@ima.umn.edu). Supported, in part by grants EP/C000285 and NSF-DMS0439734. S. B. Damelin thanks the Institute for Mathematics and Applications for their hospitality.
Department of Mathematical Sciences, Indiana-Purdue University, Fort Wayne, IN 46805
(dragnevp@ipfw.edu).
27
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28
D. BENKO, S. B. DAMELIN, AND P. D. DRAGNEV
`<
:<
of the equilibrium measure
is a major step in
The determination of the support
obtaining the measure. As described by Deift [8, Chapter 6], information that the support
consists of
disjoint closed intervals, allows one to set up a system of equations for
the endpoints, from which the endpoints may be calculated. Knowing the endpoints, the
equilibrium measure may be obtained from a Riemann-Hilbert problem or, equivalently, a
singular integral equation. It is for this reason that it is important to have a priori conditions
on the external field to ensure that the support is an interval or the union of a finite number of
intervals. We refer the reader to the references [3, 4, 5, 6, 9, 10, 11, 15, 17, 18] for an account
of advances on the equilibrium measure and support problem for one or several intervals.
In this present paper, we study supports of equilibrium measures for a general class of
weights on the compactified real line and unit circle and present several conditions on the
associated external field to ensure that the support of the associated equilibrium measure is
one interval or one arc.
In order to present our main results, we find it convenient to introduce some needed
notation and definitions.
denote the compactified real line. It is a topological
D EFINITION 1.1. Let
space which is isomorphic to the unit circle . We will think of
as
, that is, we agree
that
for any
.
Let
. Then
denotes an interval which is open,
closed, or half open, and has endpoints and . We define
,
,
,
.
Let now
be two angles,
We define
to be the arc
, where we go from
to
in a counterclockwise direction. If
let
to be the full circle C. If
or
, then let
be the single point
. Finally, if
and
then define to be
.
We say that
,
is a weight on , if
acb H
e d f he g i* j9
k
l)
e
mon
o
m
.
; vuNpZqxwU e d
pZqr. e d prstq
p q
y q>ph
z[ph{q2%}| 3q>p~%2
yfpZ{qC| [qN}p"
xYpZq€e | y q>p)%Z
\yfpZq2I %| I
‚>ƒ„.
ƒ\‚ n†…&‡1O
ˆu=‚(#ƒTw
u=‰‹ŠŒŽ{‰-Š!w~jk
‰‹ŠŒ ‰-Š
ƒWL‚L‘…&‡1
ˆu=‚(#ƒŽw
‚0UƒL‘…&‡1 ‚’jƒ
;™ ˆuP‚>ƒŽw ˆu=‚(#ƒŽw
!"P“”‚•%
7–s—ƒ˜’‚e s8…&‡ ; tuPe ‚>ƒŽw
š›=œL% œ. d
d
(1.1)
2PžŸ%$
f š›I H ‹L ¡T£¤ž ¢¢ “I % I ž I H
is a weight on k .
R
1. We note that this definition of weights on the real line is B more
I œ general
I š›PœL%
¥
B
¦
D
F
¥
R
Q
thanI the
one
given
in
[17]
or
[18],
since
we
do
not
assume
the
existence
of
I
I
I
=€% is bounded from below, œ š›=œU% must be
as œ „ . However, since „
EMARK
bounded from above. In addition, studying weights on the compactified real line via weights
on the unit circle allows us to deduce several results on the supports of the equilibrium
on the line via a general result for
on the circle (see Theorems 2.1, 2.3 and
measure
2.4).
In the next subsection, we describe the relation between the weighted energy problem on
and on .
:¨§
ed
k
k
:<
ed
1.2. Connection between the equilibrium problem on
use of the Cayley transform between and as follows.
k
e‘d © œ«ª¤¬ž0
z
œ L“ .­k
œ l“
t
e
defines a bijection between d and C. The inverse is
H
k © ž­ªŸ¬œ® H l¯Užž “(. e d O
ed
and on
k
. We will make
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SUPPORT OF EQUILIBRIUM MEASURE
29
°>±². e d by thee Cayley transform will be denoted by ³ and K .
:L.´VX d % , we assign the Borel probability measure :¨µ on k with
M :Tµ$=ž¤%Z
M :N=œL%O
e
This mapping is a bijection between Borel probability measures on d and k .
Let the weights š and be related by (1.1). The weighted logarithmic potential of :
and :Tµ is defined by
? B¥D¦F
H
I ±8Uœ I š›=±%š›=œL% M :N=±%
pC§ ¶ =œL%6
<p ¶&· =žŸ%h
f ? BEDGF I I H
K•Lž 2PK#%”2PžŸ% M : µ PK#%}
respectively ([18]). These are well-defined integrals (even though : may not have compact
support), as well as
; §’P:"%6
fx ?¸? BEDGF I œ¹° I š›=œU%#š›P°º%#% :NPœL% :NP°/%}O
M M
From
I œzU° I ¼» HH lž “" HH l³ “‹» I H … I žXIELI H ³ I I
»» Lž
L³ »» Už U³
I
I
I
I
we have ±8Uœ š›=±%š›=œL%½‘… K•Už 2=K#%2=ž¤% . Thus,
BEDGF
pC§ ¶ =œL%Np < ¶&· PžŸ%N …¾O
(1.2)
The image of
To any measure
Integrating this we get
(1.3)
; §’P:"%> ; < P : µ %• ¥B D¦F !… O
Since
šc‰ £¤¿ À²j‰ £¤Á we have the following correspondence between and  :
H
BEDGF
(1.4)
!PžŸ%>²ÂLà H l¯Užž “YÄ/l I H Už I I ž I H O
For convenience we will agree on the notations
!=Ŧ%h
f‘!P‰ ŠÆ %¨2PŦ%Z
j–P‰ ŠEÆ %Àź. e O
Also, since
we have
(1.5)
I H Už I I … I Ç H … I ž I H cœr.e d œ l¯“
t
¯l œÉÈ
U“ Ä l H BEDGF H lœ È %• ¥B D¦F !… Àœ¬. e d O
ÂPœL%Nj à œ¹
œ¸l¯“
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30
D. BENKO, S. B. DAMELIN, AND P. D. DRAGNEV
³/‘‰‹ŠÊ Å {Ë´. e
IÌ
I
I I
I H Užž UIEI H ³ L³ I … #I Ì ER SZÅ4R¥Í&S … I¥Æ I#£ È Ì REÊ S$ËÍ&… I
We find it more convenient to use angles instead of complex numbers on the unit circle.
, and
for
.
So let
Clearly,
ž´‰‹ŠEÆ
(1.6)
and
D&Ï
H l ž
H Lž ½“ \­Î Ņ O
Therefore, using (1.6), we readily calculate that
; § =:"%N
? ? BEDGF Ì Å)’Ë š›˜Î D&Ï Å4ÍG…G% š›#­Î D&Ï ËÍG…G%
t
x
Ãл» RES … »» IÌ REShÅ4Í&… I I#Ì RES^ËÍG… I Ä M :
»
»
ÑjÒ ˜Î D&Ï Å : à ˜Î DGÏ Ë Ä
?t? BEDGF Ì …TÓ0Å)M ’Ë
D&Ï Å
D&Ï Ë B¥D¦FNÔ
Ò
x
Ãл» RES … »» 2PŦ%”2[˾%Ä M : ­Î …ŽÓ M :ÉÃG­Î … ÄX O
»
» D&Ï IÌ I
Here, we used the fact that 2PŦ%$\š›˜Î
ÈÆ %{;3… R¥S ÈÆ % (see (1.1)). In addition, we note
that from (1.4) we get
DGÏ
BEDGF
BEDGF
(1.7)
!PŦ%>Â Ò ˜Î Ņ¤Ó l IÌ R¥S Ņ I l …!O
The formulae (1.1)–(1.3) allow us to conclude the following:
e
:¹.ÕVX e d % minimizes the energy integral ; § P:"% over all probability measures
on d
]
;
<
=:Tµ>% over
if and only if its corresponding :¨µ.ÕVX3k)% minimizes the energy integral
all probability measures on k . Moreover, the support `½§ is going to be an interval or a
<
e
complement of an interval in d if and only if the corresponding support ` is an arc on k .
We close this section by introducing some remaining conventions which we assume
henceforth.
;
; e is absolutely continuous inside ; Ö if
Let Ö be an arc of k . We shall say that ×
CÖ ;
it is absolutely continuous on each compact subarc of Ö . (As a consequence, ×TØ exists a.e. on
; Ö .)
;
e
;
let be an interval or a complement of an interval in d . Let the arc Ö be the image of
; byNow
e
;
e
the Cayley transform ±
d Ùk . We shall say that ×É
is absolutely continuous
;
; ;
inside if ×)ڕ± £T is absolutely continuous inside Ö O (If is a finite interval, this definition is
;
equivalent to the usual definition of absolute continuity inside .)
;
e
; such that
We say that a function × is increasing on an interval if there exist ÛU
~
;
Ü
the Lebesgue measure of
Û is zero and ×½=ž¤%~sÝ×½; =³¾% whenever ž¨#³.jÛ , ž—s\³ . (This
is a useful definition when × is defined only a.e. on .) We define “decreasing” in a similar
manner.
;
;
Moreover, we say that × is convex on an interval if × is absolutely continuous inside
;
and ×¤Ø is increasing on .
We finally note that under Cayley transform (or its inverse), sets with positive capacity
are transferred to sets with positive capacity.
The remainder of this paper is structured as follows. In Section 2, we present our main
results and in Section 3 we present our proofs.
2. Main Results: The Circle and the Compactified Real Line. In this section we state
our main results. We begin with our main results for the circle and compactified real line.
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SUPPORT OF EQUILIBRIUM MEASURE
2.1. Circle.
31
2 P,A%^+Þ!"$!3,4%%} I , I H be a weight on k and let ; ; ßuà•{á-w
7ºn8á^LàÉsj…i‡1O B Assume that is absolutely continuous inside and
(2.1)
žWREQ¬RES!; ³ â !=žŸ%(!P³!%
ž˜.
;
;
whenever ³ is an endpoint of with ³o. . Let ‰iŠ| be any point which is not an interior point
;
of ™ . Let ‚ ˆy #ƒ []OÞO]Oހ‚½
ˆy ã4#ƒŸã‹ be äWb7 arcs of ; k . Here, for all H s“>s8ä , 7/nƒ Š /‚ Š s8…&‡ H
<
g
;
and 3`
#ƒ Š . Suppose further
; ™ % ;y ˆ‚ æ Š and
H that can be written as a disjoint union of å¯b
intervals ÞO]OÞOÞ
for any fixed s’çs—å , either
D
F
(2.2)
‰ Á}è Æ{é!ê … Ì RES à ŀ’… ë Ä Ø =Ŧ%NUÎ Ì Ã Å)’… ë ÄÐì Ì S Ã Ì RES à ŀ’… ë ĎÄ
;í
H
is increasing on , or for some s—“Ns8ä :
Ì RES Ò Å)’‚ Š Ì R¥S Ò ƒ Š UÅ Ø PŦ%"l ÔH Ì R¥S Ò ÅC ‚ Š l¯ƒ Š
(2.3)
Ó
Ó
Ó
;
í
is increasing on . Finally, we assume that
B Ì#î
B
Ø
REQ ñ PŦ%Zs REQRES!ñ â Ø =Å4%}
Æï~Æò
Æï~ÆÞð
<ºô ; ™ is an arc
;í H
;
whenever Åi' is an endpoint of ( sçsóå ) but not an endpoint of . Then `
of k .
Here sgn denotes the signum function.
R
2. The choice of ë is not important, see Remark 6 and the proof of Lemma
;
3.3. We also remark that if ™ is the full circle, then one should check only condition (2.2) and
ignore (2.3) which is a stronger assumption.
Below we give a condition which guarantees that
` < isH the full circle:
I
I
; C
2.2. Let 2P,A%ºtÞA¨$!P,A%#% , be a weight on k and let
;
#à ló…i‡"% and È \à È à È ló…i‡"% where; ‰iŠõ]ö2‘÷ ‰‹Šõø . Assume that< (2.2) is increasing on
;à where
is increasing on È where ë€
—à È . Then `
k .
<~ô ; ™ is an arc of k . Let ‰ Š| be an interior point
Proof.ë€
By8à Theorem
, and (2.2)
2.1 `
of this arc, not
identical to ‰‹Š¥õ ø . Choose ù ù È such that ë)n8ù È njù një>l—…i‡ and both of the arcs 3ˆ ëi#ù %
and =ù È ˆ {ë1l…i‡"% contain only one of ‰ Š¥õ]ö and ‰ Šõø , say, Pˆ ëiù % contains ‰ Š¥õ]ö and =ù È ˆ ë>l…i‡"%
contains ‰‹Š¥õø .
Using the first observation of Remark 2, we see that (2.2) is increasing on 3ëi#ù % because
(2.2) is increasing on 3ëi#ù % when at (2.2) ë is replaced by à È . Similarly, (2.2) is increasing
on Pù È ë>l…i‡"% because (2.2) is increasing on Pù È ëNló…&‡"% when at (2.2) ë is replaced by à .
< k by Theorem 2.1 and by the choice of Thus (2.2) is increasing on Pëi{ë>l…&‡"% and so `
ë.
Ô
Example. TheD following
example illustrates the theorem.
Ì
Ì
g
]
H
ý
Let !PŦ%2¼Î [úiŦ% RES"PûGŦ% defined on Ý«y …!O ü¾{û¾O y ûO ü4ú! . < (Weô may define
H
ý
k
`
y
¾
ˆ
O
ü

û
O
to be zero outside
so
that
is
defined
on
.)
We
claim
that
both
and
Ô
` <Wô û¾ˆy O ü¦ú¾ are arcs of k . (One of them may be an empty set.)
T HEOREM 2.1. Let
be an interval with
EMARK
OROLLARY
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32
Ô
D. BENKO, S. B. DAMELIN, AND P. D. DRAGNEV
‚È
ûO ü4ú!ƒ È ûO H-H‹ý þ ló…&‡
y …!O ü¾{û¾O ÔH
‰ ŠE|
3&؍% È ljiØ ØÐl Í bÝ7
y …!O ü¾{û¾O H]ý i
H
þ
H
ý
y ûO´< ô  ûO H]ý
` y …!ˆO ü¾{û¾O ‚ …!O ü¾ƒ y …!O ü¾{û¾O H]H]ý ý y …!ˆO ü¾{û¾O ‚ ƒ and
.
Take
One can verify that (2.2) is satisfied on
but not on the whole
. (At
(2.2) can be chosen to be any number such that
is not an interior point of
. Or,
simply check the
condition, see Remark 6.) Also, using
and
we see that (2.3) is not satisfied on the whole
. However, (2.3) is satisfied on the
subinterval
(see F IG . 2.1). So the combination of the (2.2) and (2.3) conditions
implies that
is an arc.
ë
Condition (2.2) on [2.9,3.18]
Condition (2.3) on [2.9,3.18]
6
0.05
4
x
0
2
0
2.95
3
3.05
3.1
3.15
-0.05
2.95
3
3.05
3.1
3.15
x
-2
-0.1
Ô
ÿ
F IG . 2.1. Conditions (2.2) and (2.3) on the interval ‚ ƒ Ô
y û¾O ü¦ú¾ Ô
y û¾O ü¦ú¾ Ô
`H]ý <´ô û¾ˆy O ü¦ú¾ y …!O ü¾{û¾O Using
and
on
is not helpful since (2.3) is a decreasing function there.
Also, (2.2) is not satisfied on the whole
. However, (2.3) is satisfied using
and
on the whole
. Theorem 2.1 now implies that
is an arc (see F IG . 2.2).
and
are not helpful on
since (2.3) is a decreasing function
(We remark that
on
.)
y û¾O ü¦ú! ‚ ƒ
y ûO Hiþ ûO H-ý È È
Condition (2.2) on [3.95,4]
‚È
ƒÈ
Condition (2.3) on [3.95,4]
x
3.95
3.96
3.97
3.98
3.99
5.45
5.4
-0.075
5.35
-0.08
5.3
-0.085
5.25
5.2
-0.09
5.15
3.95
3.96
3.97
3.98
3.99
x
ÿ
F IG . 2.2. Conditions (2.2) and (2.3) on the interval R EMARK 3. It is a natural question to ask what
‚ Š and ƒ Š numbers we should choose in
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33
SUPPORT OF EQUILIBRIUM MEASURE
order that (2.3) is as weak as possible. In most cases the following statement is true:
Let
and
(
) be two arcs of such that
. Let be an arc contained in
. If (2.2) or (2.3) is satisfied with
then (2.2) or (2.3) is also satisfied with
.
For example, this statement is true if
exists and the sets
y ‚>ƒŽ y ˆ ‚ Ø ƒ Ø ` < y ‚(#ƒŽN y ˆ ‚ Ø ƒ Ø ‚¨Ø[ƒ"Ø
Å/. ; G Ø =Ŧ%h5
7/; n—ƒ˜’‚ós…i‡1½7/n—ƒ¨Ø¾’‚¨Ø¨s…i‡
k
™
y ‚(#ƒT
‚>ƒ
GØ ØYPŦ%
H D&Ï Å)’‚¨Ø
H DGÏ
΅ à … Ä f Å. ; G Ø =Ŧ%Z5 … Î Ã Å)L … ƒ¨Ø Ä
consist of finitely many intervals. (The proof of this is similar to the proof of [3, second
remark].)
Theorem 2.1 can be effectively used when
is identically zero on some arcs (that
is, is a subset of finitely many arcs). If
is zero on ( ,
, then we may choose
to be in Theorem 2.1. This is consistent
with the discussion above. For convenience we will state Theorem 2.3 in accordance with this
remark.
2 3,4%
2P,A%
y ˆ‚ Š #ƒ Š y ˆ Š Š “) H Þ OÞO]OÞä
y ˆ Š Š 7n Š Š n…i‡"%
2.2. Compactified Real Line.
äo. ¡ let
g
f Šã y Š Š ¨ e d À Љ i‰
n s n È s È n ¾n ã–s Cã2njl)O
; be an interval and assume that  is absolutely
Let š ‘!T#^Â~% be a weight on , ;
continuous inside and
B
(2.4)
œ®REQ—rR¥S!;° â ÂPœL%1‘ÂoP°–%
œ¬.
;
;
;
whenever ° is an endpoint of with °Ý. . Assume further that can be written as a disjoint
;
;
æ
H
union of intervals ÞO]OÞOÞ
such that for any fixed sçså either
‰ ¿1è é is convex on ; í H
H
or for some s“>s8ä–
=œ« Š % Š ¡¨ UœL%#Â Ø PœL%Tlœ is decreasing on ;í or
=œz %Þ ã UœL%Â Ø PœL%Tl¯œ is increasing on ; í O
(2.5)
Finally, we assume that
B Ì#î
B
REQ ñ Â Ø PœL%6s REQRES!ñ â Â Ø =œU%}
ï ò
ï í Hð
ô ; is an
;
;
whenever œ ' is an endpoint of sóçWsjå"% but not an endpoint of . Then ` §
interval.
R
4. We remark that Theorem 2.3 is also valid when one interval, say, y CãA Cã-
is an infinite interval or a complement of a finite interval. If )ã‘5 Cã (and, of course,
€ã–n ), then the conclusion of the theorem holds if (2.5) is replaced by the condition:
(2.6)
=œz CãG% UœL%#Â Ø =œU%Tlœ is decreasing on ;í O
T HEOREM 2.3. For given
!
EMARK
#
!
"
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34
D. BENKO, S. B. DAMELIN, AND P. D. DRAGNEV
If however ã ²l)
, then (2.5) should be replaced by the condition:
=œz %#Â Ø =œL% is increasing on ;í O
Finally, if
¼ (and so y is the infinite interval instead of y Cã! Cã] ) then (2.5)
should be replaced by the condition
㠒œL%Â Ø PœL% is increasing on ; í O
;
At (2.6) and at Theorem 2.4 at (e) one can also consider an which is a complement of
a bounded interval. We leave the details for the reader.
Theorem 2.4 reveals to us the following remarkable connection between previously known
conditions on . It also gives us a new condition (which is (e) below). As a consequence of
Theorem 2.1 and 2.3 and Remark 4, we now have the following general result for the case
! when is one real interval. See also [3]. Recall that for we define $
.
be a weight on and let
be an interval. Assume that is
T HEOREM 2.4. Let
absolutely continuous inside and satisfies (2.4). Let % be finite constants and suppose
that either of the following conditions below hold:
$ ,
$ .
(a)
is increasing on
(b)
is increasing on
,
.
$ ,
(c) !
is increasing on
.
(d)
is increasing on
,
(e)
is decreasing on
,
.
(f)
is convex on .
is convex on .
(g)
Then
is an interval.
R EMARK 5. Theoretically one should ignore (d) and (f) since (g) is a weaker assumption
than both of these. Nevertheless we included them here, because sometimes they are easier to
check.
Notice that (a) in Theorem 2.4 corresponds to the case of Theorem 2.1
when
is
an arc of disjoint of the point
, (b) corresponds to the case when
is a proper
subarc of such that
, (c) corresponds to the case when
is
a
proper
subarc
of such that
, (d) corresponds to the case when
is the full circle and
a subcase of this is when (so
and
)
which
corresponds
to (f). The
condition (e) corresponds to the case when
is a proper subarc of which contains the
point
inside the arc. Finally, (g) is the only condition which corresponds to (2.2) and
not (2.3).
Note also that if we let & then (e) leads to condition (d), since
'
#
is decreasing if and only if
is increasing.
One may also combine the above conditions to create a weaker condition in the spirit of
Theorem 2.1 and 2.3.
Â
y o h•
f 2 –%{|
e
; +e n
š ;
Â
\s
; ²y ~ ) ` § ²y ~ )
=œz % \’œL%#Â Ø PœL%Tlœ ;
y ~}l)ó% `¨§®y ~}l)ó%
=œz %#Â~ØY=œL%
;
#; j e) Ü `¨§ß² ~
\UœL%Â)ØPPœL%
=œz % È Â)Ø[=œU%•Lœ
; * )9
=œz % \’œL; %#Â~ØPPœL%Tlœ
+y o $ `¨§«+y o $
Â
;
!"[Â~ô % ;
`¨§
y ‚(#ƒT
H
k
ž
y ‚(#ƒŽ
k
!"P“[H ƒ•%N H
y ‚>ƒŽ
k
!"P“”‚•%$
k
y ‚(#ƒŽ
¹z ‚+ 7 ƒ ¹…i‡
y ‚(#ƒT
k
žL H
Ù
= œ„ % œL%Â)Ø3PœL%Tl¯œ
Pœz €% È Â)Ø[=œL%•Uœ
3. Proofs. In this section, we present the proofs of our results. We find it convenient to
break down our proofs into several auxiliary lemmas. Our first lemma is
L EMMA 3.1. Let
,
be a weight on and let
be
an interval with
. Let
and assume
. Suppose
is absolutely continuous inside and satisfies (2.1). Moreover, assume that
2 3,4%hÞA¨$!P,A%#% I , I H
7/n—á(Wà0s8…&‡ 7/n󃭂; s8…&‡
!=Å4%Z
f‘!P‰‹ŠÆ]%
Ì R¥S Ò Å)U‚ Ì R¥S Ò ƒ˜UÅ Ø PŦ%¨l ÔH Ì R¥S Ò €Å (3.1)
Ó
Ó
k< g ; ; «
uà•{á-w
` ™ y ‚(#ƒT
‚´l¯ƒ
Ó
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35
SUPPORT OF EQUILIBRIUM MEASURE
` < ô ; ™ is an arc of C.
DGÏ ‚
D&Ï ƒ
DGÏ Å
+
\˜Î … ›
x­Î … Àœ¬
\˜Î … O
First, let us assume that ‚>ƒ.jP7¾…i‡"% . Thus, we may assume that 70n‚²s‘à—n²Å˜n
á2s—ƒ’n8…&‡ and 7ºn Ì R¥S¨P‚1Í&…G% , 7ºn Ì RES"=ƒ•ÍG…G% . So +sœrs .
From (1.7), we have
D&Ï Å
H D&Ï Å
Ì
Å
Ò
Ò
Ò
(3.2)
Â Ø ˜Î …¤Ó ²… RES È …TÓ Ø =Ŧ%N … Î …ŽÓ O
;
is increasing on . Then
Proof. Let
(
%
Thus,
= œz D&%ÞÏ \UœLDG%#Ï Â Ø PœL%TlD&Ï œ D&Ï
DGÏ
+ Ò Î Å… ’Î ‚ …1Ó Ò Î Å… UÎ ƒ …½Ó Â Ø Ò ˜Î ŅŽÓ
Ì RES Æ £ Œ Ì R¥S Æ £  Ò
D&Ï
D&Ï
+ Ì RES È Œ Ì R¥S  È …G Ø PŦ%½’Î Ņ¤Ó UΠŅ O
È È
Now we use the following identity which holds for any ‚>ƒN#Å :
DGÏ Å Ì RES Æ £ Œ Ì RES Æ £  H Ì RES"=Å) Œ ¡  % H Ò DGÏ
Î Ã … Ä$Ã Ì RES È Œ Ì RES  È Ä … Ì REST Œ % Ì R¥S"È  % … Î
È È
È È
!
It follows that
D&Ï
UÎ Å
‚ ól Î DGÏ ƒ O
1Ó
P œ¹ Ì%Þ LÌ œL%Â Ø PœL%Tl¯œ Ì
H DGÏ
D&Ï
x^… R¥S Ì R¥Æ S £ È ŒŒ Ì RERESS Æ £ È  Ø =Ŧ%"l … Ì R¥S¨R¥ST=Å)Œ % Ì Œ RES"¡ È   % % … Ò Î ‚ … lóÎ ƒ …NÓ O
È È Ì
È È
Ì
;
Because 7˜n RES"P‚1Í&…¦% , 7˜n R¥S¨=ƒ•ÍG…G% , the right hand side of (3.3) is increasing on if
DGÏ if (3.1)D&Ï holds. Thus, if (3.1) holds then Pœ %Þ ’ºœL%#Â~Ø[=œL%&loœ ise increasing on
and only
u”˜Î õÈ ]˜Î È w . Now consider the corresponding equilibrium problem on d e , as described
D&Ï corresponding
D&Ï equilibrium measure on d . < Using
in Section 1 and let ` § denote
the
[3,
ô
u­Î õÈ Þ˜Î È w is an interval. It follows that ` ô ; ™ is an
Theorem 7] we get that ` §
arc of k . This proves Lemma 3.1 for the case when ‚(#ƒ’Ì .P7¾…i‡"% .
Ì
Now let ‚8s²…&‡ósƒNƒ0¯‚8n²…&‡ . Note that 7Ws R¥ST3‚½ÍG…G%}{7´b RES"Pƒ•Í&…¦% . We cannot
apply [3, Theorem 7] because Ùs
and œ is outside y o h . However, we can use the
observation that condition (3.1) is “rotation invariant.”
be a number such that
Let 7/n
7/n‚É 0–
&‚ Àƒ fƒ˜ Énj…i‡1
(3.3)
!
+*
)
!
+*
,
-
#-
and define
-
à fjàX ¤•á ‘á ¤
-
-
È =Ŧ%Z
‘!=Ål T%}O
.-
$
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36
D. BENKO, S. B. DAMELIN, AND P. D. DRAGNEV
È
!T< $ô È %
` ø àˆu á w
` < ô ;™
ƒ0¯‚8
x…&‡1O
‚ #ƒ #à á
and the parameters
, we may apply the case we studied
For
above to get that
is an arc of C. But this new equilibrium problem is isomorphic
to the original one in the sense that everything (including the support) is rotated by the angle
- . It follows that
is an arc of C.
Finally, we need to establish the lemma for the case when is the full circle. So let
Using the rotation invariance we may assume that
. Condition
(3.1) is now equivalent to
;™
‚\7¾#ƒ¯\…&‡
Ì RES È Ã Å Ä Ø PŦ%½ ÔH Ì ER ShÅ is increasing.
Using (3.2) we get
H
DGÏ
Ì RES È Å… % Ø =Ŧ%1 … Ì R¥ShŖ‘Â Ø ˜Î Ņ %}O
(3.4)
DGÏ
Thus, Â)Ø[˜Î
that is, Â~Ø3PœL% is increasing, and so Âo=œL% is
ÈÆ % is increasing (7onÅ´n²…&‡ ),case
` § is an interval. (The proof
convex. It is well known, see [17], that in this
< is againtheansupport
works for our more general weight.) So `
arc. We have completed the proof
Lemma 3.1.
As a corollary to Lemma 3.1, we have
e
L
3.2. Let š be a weight on d , let Û be a finite interval and suppose that  is
absolutely continuous inside Û and satisfies condition (2.4). Let ¹s
be finite constants
with ÛUy o h , `¨§«+y X $ and assume that Pœ €% +0œL%#Â~Ø[=œU%Ÿlœ is decreasing
ô Û is an interval.
on Û . Then `"§
D&Ï
D&Ï
Proof. Recall that y o hŽx 2 –% | , see Definition
1.1.
We mayÌ find ‚Õn¼
r®˜Î 3‚½ÍG…G% , «ß˜Î =ƒ•ÍG…G% and ƒ’—‚Ýs¸…i‡ .
Ì ƒ such that
Ï 7 necessarily.
Notice that R¥S~3‚½ÍGD&…GÏ % R¥S~Pƒ•Í&…¦%6D&n
Let Ûɸu”˜Î =àTÍG…G%}]˜Î 3áGÍ&…¦%Yw , where ‚sàÉsá–s—ƒ and so á€ÉàÉs8…i‡ .
The left hand side of (3.3) is a decreasing function of œ on Û , and so the right hand
;
side of (3.3) is a decreasing
function
of Å on fy à•{á- . Multiply that right hand side by
Ì
Ì
the negative constant R¥S~3‚½ÍG…G% R¥S)Pƒ•Í&…¦% . In this way we get an increasing function of Å on
y à•{á- . So condition (3.1) is satisfied and fromô Lemma 3.1, we deduce that ` <˜ô y ཁá- is an
Û is an interval. Lemma 3.2 is proved.
arc of C. This implies immediately that ` §
Our final lemma is:
I I H be a weight on k and let ; ßuà½á-w be
L
3.3. Let 2P,A%$ÞA¨$!P,A%#% , ;
an interval with 7­n+á)’àós\…i‡ . Suppose is absolutely continuous inside and satisfies
;
(2.1). Let ‰‹Š| be any point which is not an interior point of ™ . If
D Ì Å)’ë Ì F Ì Å)Uë
Ì
)
Å
U
ë
‰ Á}è Æ{é ê … RESXà … Ď Ø =Å4%1’Î à … ğì S(à RESXà … ĎÄ
(3.5)
<´ô ;™ is an arc of k .
;
is increasing on , then `
;
R
6. Whether (3.5) is increasing on or not, it does not depend on the choice of
;
ë (as long as ‰ ŠE| is not an interior point of ™ ). The proof of this is given in the proof of Lemma
3.3. We also remark that if is twice differentiable then condition (3.5) is easily seen to be
equivalent to
H
Ø =Ŧ% È ló Ø Ø PŦ%¨l Ô b—7 Å.’ཁá&%
(regardless of the value of ë ).
EMMA
$
$
0
! /
EMMA
EMARK
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37
SUPPORT OF EQUILIBRIUM MEASURE
We give the following example to Lemma 3.3. Let be one or several closed arcs on the
unit circle but not the full circle. Assume the weight is zero on the complement of . Let
21 be a point in the complement of , and define
EB DGF #I Ì Å)Lù I
!PŦ%Z
!3‰ ŠEÆ %6
f
ER S … l M where is an arbitrary constant. The value of ë is our choice so let ë­
¸ù . Then (3.5) is
< is a set of arcs.
M
increasing on the whole of (in fact it is identically zero)
and therefore `
<
Moreover, each arc of contains at most one arc of ` .
;
Proof of Lemma 3.3. First we show that whether (3.5) is increasing on or not, it does
not depend on the choice of ë . We do not assume the existence of 4Ø Ø .
H
Let =žŸ% and •PžŸ% be two real functions on 37¾ % such that is bounded and increasing,
H
and is non-negative and Lipschitz continuous. Then there exists Ý+P7¾ % of full measure
such that
?
à =ž¤% ½=ž¤% Ä Ø M žÉs² Ž% %½8 Ž%3mA% if mÐ . ½mos ‹O
This observation easily follows from Fatou’s Lemma applied to the sequence of functions
y¥ ¤%Þ=žºl æ %½8 Ž%=ž¤%YPÍ æ , æ r7 ¡ .
;
;
Suppose ‰‹Š| and ‰‹ŠE| ø are not interior points of ™ . Denote now (3.5) by =Ŧ% . Let ÛU
|
such that Û has full measure and PžŸ%´s
|
| =³¾% for all ž+s¸³ , ž"#³‘.xÛ . We define the
domain of
and iØ to be Û . We have
|
D Ì ÌF Ì
=Ŧ%"ló‰ Áè Æ{é Ã Î Æ £È | Ä S à RES Æ £È | Ä
(3.6)
‰ Áè Æ{é Ø =Ŧ%( |
ÀÅ.LÛ¾
»» Ì RES Æ £È | »»
»
»
which shows that ‰ Á iØ is differentiable a.e. on Û . Simple calculation gives
H
Ì
)
Å
U
ë
Ô
Ø
ê
{
Æ
é
Ø
Ø
(3.7)
7ºs | =Ŧ%(‘…T»» RES … »» P‰ Á}è PŦ%#% l ‰ Áè Æ{é ì a.e. Å/.LÛ¾O
»
»
Replace ë by ë È at the formula (3.5) and denote it by
replace in that formula
| ø PŦ% ½. =Also,
‰ Á iØ by the quotient at (3.6). Thus we see that with some
Ŧ% , ¤=Å4% functions |Yø =Å4%X
holds, where inside ཁá&% : the function is non-negative and Lipschitz
| PŦ% ½=Ŧ%6l ¤=isÅ4% increasing
continuous,
; and bounded, and is absolutely continuous (since ‰ Á is abso|
lutely continuous inside ).
‰‹Š
3
3
4
%5
3
6
38
.<
!38
7
!38
=<
:9
!38
9 ;4
9
<
3
3
>3
3
3
3
3
3
7
3
3
So, by the observation above, we have
?
%5
| /l !% Ø s² | Ž% Þ%Tl ¤ %1— | Ž%3mA%• ¤Pm!%> |Yø %½ |Yø Pm!%
;
;
for a.e. mŸ X. , where m¯s
. But this integral is non-negative, since 7Ls –Ø a.e. ÅL.
|
ø
follows from (3.7). Hence, 7s
| ø %½ | ø Pm!% , i.e., | ø is increasing. And this is what we
wanted to show.
;
We may assume that ësà—n+á­s\ëZlj…i‡ . Let us rotate now ™ to a position such that
H
the rotation takes ‰iŠE| to the point žó
. Condition (3.5) will change accordingly to a new
condition where now ë€+7 . (We denote the new rotated weight by x²!"#$&% , too.) We
<´ô ;™ is an arc of k for the new ` < and new ;™ . Once we have done
now have to show that `
;
that we simply rotate ™ back to the original position and the proof is complete.
6
$9
3
?
!3
@9
3
9
?
9
3
#
3
:9
3
,3
:9
3
3
ETNA
Kent State University
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38
D. BENKO, S. B. DAMELIN, AND P. D. DRAGNEV
This argument shows that we can assume without loss of generality that
7/s¯àLnáºs—…&‡1O Define
(3.8)
H
š à H l¯U žž “ Ä f I H Už I 2PžŸ%} I ž I H O
ëó 7
and
Using the arguments in Section 1.1, (3.8) may also be given as
& £ Š
š =œL%(
Ç H à l¯Z¡œÉŠ ÈÄ Àœ. e d O
›
We define Âo=œL% by š›PœL%N–
GÞ!"^ÂPœL%#% . Since is a weight on k , we know that š is
e
a weight on d .
We now show that ‰ ¿1è é Â Ø =œU% is increasing on
; ' ¸u”˜Î D&Ï à ޘΠDGÏ á w¦O
Let ž´‘‰‹ŠEÆ&O Note that from (1.7) we have
‰ ¿Nè é … I#‰ Ì ÁREèS Æ{é Æ I O
È
Using this and (3.2), for ź.’y 7¾{…&‡¤ we get
H
D
‰ ¿1è é Â Ø =œU%N … ‰ Á}è Æ{é 3… Ì R¥S Ņ Ø PŦ%½’Î Ì Å… %}O
(3.9)
;
Note that the right hand side of (3.9) is an increasing
function of Å on by assumption. Now
ô
;
B that ` § ' is an interval. (Although this theorem
we apply [3, Theorem 5], to conclude
is formulated for weights with R¥Q
ï œLš›PœL%C¼7 , the argument in the proof may be
applied word for word for the more general weights considered here. Naturally one should
ô ; ' is an interval, we conclude that ` <­ô ;™ is
work with p)§ ¶ PœL% in the proof.) Since `¨§
an arc of k . The proof of Lemma 3.3 is complete.
We are now ready to present the
;
Proof of Theorem 2.1. If ™ is the full circle k , then it follows from the assumption that
‰‹ŠEŒ Ʉ‰-Š ÉÙ
‰‹ŠE|’Ù‰‹Š¥õ for all K . Now, if (2.3) is increasing on ;í , then (2.2) is also
;
í
increasing on , as one can see. (Choose à to be zero and use (3.4), (3.9) and the fact that
the convexity of  implies the convexity of ÞA¨3Â~; % .) So we can get the weakest assumption
if we assume that (2.2) is increasing on the whole , and we already know from Lemma 3.3
;
that Theorem 2.1 holds under such an assumption. Thus, let us assume that ™ is not the full
2
2
2
0A
A
CB
ED
GF
=F
circle.
As in the proof of Lemma 3.1 and 3.3 we observe that the statement of Theorem 2.1
is “rotation invariant.” So, we may assume that
does not contain the
point and
, HF
for any . We can also assume that
.
GF
Let
,
, and
be defined by
I
(1.5). Let be given by
y ཁá-
D&Ï
‰‹ŠEŒ € ÷ H ‹‰ Š ÷ H DGÏ K
ë$‘DGÏ7
œ ;í ˜Î =ÅAÍ&…G% Š ˜Î 3‚ Š Í&…¦% Š ˜Î =ƒ Š ÍG…G%
;{í ¸u í í w4 7/n í í n—…&‡1
KJ
and define
L
.L
ž H
Âo=œL%
#J
; í ' ¸u­Î D&Ï í ] ˜Î D&Ï í w¦ ; ' f
u”˜Î DGÏ à ]˜Î D&Ï á w¦O
J
L
ETNA
Kent State University
etna@mcs.kent.edu
;'
H ]OÞOÞOÞ#å
39
SUPPORT OF EQUILIBRIUM MEASURE
e
; í'
ç/ H sUçoså"%
; í ' 2ç y Š Š ; í'
Note that is a finite subinterval of and it is the disjoint union of the intervals (
). We assume that
is numerated from left to right. Note also that $ NM
(recall Definition 1.1).
By assumption, for any
, we can find
such that either
‰ ¿1è é
(3.10)
(3.11)
(3.12)
is convex on
2
Šn Š
Šb Š
%
and
and
“N H s—“>s8ä¾%}
; í' or
Pœ¹ Š Þ% Š LœU%#Â Ø =œL%Tl¯œ
is increasing on
!
=œz Š % Š UœL%#Â Ø PœL%Tlœ
#
; í'
is decreasing on
!
or
; í' O
((3.10) is coming from the argument in Lemma 3.3, (3.11) is from Lemma 3.1, and (3.12) is
from Lemma 3.2.)
Let 4
. We can find positive constants 4
(uniquely) such that the fol$4
lowing function is a positive continuous function inside . For
,
let
H
×
È ÞO]OÞOÞ ; '
æ
ž. ; í ' E çU H ]OÞO]O#å"%
; í'
if 3.10 % is satisfied on
ã•!"3…¦Â=œU%#%
ã =œz Š % Lœ Š % if 3.11% is satisfied on ;; íí ''
×½PžŸ%Z
ã!=œz Š %=œ« Š % if 3.12% is satisfied on O
D&Ï can use the argumentD&Ï in [3, Theorem 12] to deduce the result.
Let š fÞA¨^Â~% . We
For this purpose let ²\­Î 3‚½ÍG…G% and x­Î Pƒ•Í&…¦% be any two numbers such that +n
; ' , ~ º% ô `¨§¹ . Let : fj: < »
, y ~ )"
»» è Œ ¡ ¦é È è Œ ¡ ¦é È ¡ , : È :´: . Using
<p ¶ PžŸ%CÕp < ¶ ö =ž¤%½ljp < ¶ ø =ž¤% and the monotone
theorem it easily follows that
<p ¶ PžŸ% is absolutely continuous on y ‚>ƒŽ , and so convergence
by (1.2) pC§ ¶ PœL% is absolutely continuous
on y ~ ) . Also, as in [3] one can verify that
×½=œL% Mœ YpC§ ¶ =œL%%
M
ô y 2 ) is an interval. It
is strictly increasing on y ~ ~ . By [3, Lemma 4] we get that ` §
ô
ô
;
<
;
' is also an interval and ` ™ is an arc of k .
follows that ` §
QR
PO
4
4
?
4
?
!
$
S
=[
U
!V T W
!V YXHZ
ED
[
$
ED
We conclude this section with
The Proof of Theorem 2.3 and Theorem 2.4. These follow easily using Theorem 2.1,
Lemma 3.2 and the discussion in Section 1.
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Kent State University
etna@mcs.kent.edu
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D. BENKO, S. B. DAMELIN, AND P. D. DRAGNEV
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