Computing the equilibrium measure of a system of intervals converging to a Cantor set

0.42 MB·English

by  Giorgio Mantica

Computing the equilibrium measure of a system of intervals
converging to a Cantor set
Giorgio Mantica
Center for Non-linear and Complex Systems,
Dipartimento di Scienze ed Alta Tecnologia,
Universit`a dell’ Insubria,
via Valleggio 11, 22100 Como, Italy.
3 Also at I.N.F.N. sezione di Milano and CNISM unit`a di Como.
1
0
2
Abstract
n
a We describe a numerical technique to compute the equilibrium measure, in logarithmic
J potential theory, living on the attractor of Iterated Function Systems composed of one-

9 dimensional affinemaps. Thismeasure isobtained as thelimit ofa sequenceof equilibrium
measures on finite unions of intervals. Although these latter are known analytically, their
] computationrequirestheevaluationofanumberofintegralsandthesolutionofanon-linear
A
set of equations. We unveil the potential numerical dangers hiding in these problems and
N
we propose detailed solutions to all of them. Convergence of the procedure is illustrated in
. specific examples and is gauged by computingthe electrostatic potential.
h
t Keywords: Iterated Function Systems – Equilibrium Measure – Potential Theory
a
m
[
In memory of Professor Don Luigi Verga

1
v
4 1 Introduction
1
8
Suppose that a finite amount of “charge” is placed on a set of “conductors” of arbitrary shape,
1
placedatequallyarbitrarypositionsinspace,andconnectedbythin,conductingwires. Then,this
.
1
chargedistributesitselfsotominimizerepulsion,i.e.,theelectrostaticenergyofitsconfiguration.
0
It is very easy for us to intuitively perceive this phenomenon, and indeed we have been exposed
3
1 to it since early in school, via easily realizable laboratory experiments that go back to the years
: of Alessandro Volta and to his electrophorus [1], which he did not invent, but he perfected into
v
aninstrumenttoaccumulateandtransferelectriccharge,yearsbeforehismonumentalPila. This
i
X intuitionis commonknowledgeamongscientistsofalldisciplines: inItaly,atleastuntilrecently,
r electrostatics was taught in high school, both scientific and of classical kind and, even more
a
amazingly, it was included in the syllabus of the degree in mathematics and of medical school
alike! This paper is dedicated to a wonderful teacher who let middle–school students, twelve
years–old, play with electrostatic devices and inspired them to pursue scientific research.
What is far less known is the fact that the mathematical scheme behind this physical de-
scriptioncanbeeasilygeneralizedtospacesandconductorsofarbitrarydimension—evenfractal
[15, 16]—and to arbitrary analytical forms of the “electrostatic” interaction, that need not be
“Coulomb”, but can take on different analytical forms. These generalizations have proven to be
extremelyfruitfulinawidevarietyoffields,fromPDE’stoharmonicanalysis,tonamejustafew.
In this paper, we will put ourselvesin the situation where the ambient space is two–dimensional
(the complex plane) and the pseudo–Coulomb law depends on the inverse of the distance be-
tween two charges: physicists will easily think of the interaction of two infinite, parallel wires,
mathematicianswillimmediatelyrecognizethelogarithmicpotentialthattakesusintotherealm
of the well established logarithmic potential theory [7, 10].
1
Indeed, if we let E to be a compact domain in the complex plane C, and σ a positive
probability measure supported on E, the potential V(σ;z) generated by σ at the point z in C,
is given by the formula
V(σ;z):= log z s dσ(s). (1)
−Z | − |
E
The electrostatic energy (σ) of the distribution σ is then given by the integral
E
(σ):= V(σ;u)dσ(u)= log u s dσ(s)dσ(u). (2)
E Z −Z Z | − |
E E E
The equilibrium measure σ associated with the compact domain E is the unique measure that
E
minimizes the energy (σ), when this latter is not identically infinite. Potential theory gives us
E
a formidable set of technical instruments [7, 10] to deal with this problem in relation to general
compact sets E.
Inthispaper,weareinterestedinaspecificfamilyofcompactsetsE,thatareCantorsetson
the real line associatedwith the constructionof Iterated Function Systems (IFS)—to be defined
momentarily—andwestudytheefficientnumericalcomputationoftheirequilibriummeasureσ
E
andofitspotentialV(σ;z). Ourresultswillconsistofareliablealgorithmstoachievethesegoals.
This algorithm is rather straightforward, in the sense that it combines well known theoretical
facts(theanalyticalformoftheequilibriummeasurewhenE isacollectionofintervals)andwell
known numerical techniques (Gaussian summation, root finding) with a standard idea in IFS
construction: the hierarchical approach. Despite or rather because of this simplicity, numerical
experiments indicate that the algorithm is stable and it can be used as a microscope to probe
deeply into the structure of the equilibrium measure on Cantor sets.
This paper is organized as follows: in the next section we define the attractors of Iterated
Function Systems, via a hierarchical construction. Each step in this construction yields a finite
family of intervals, whose equilibrium measure is analytically known. We sketch this solution in
Sect. 3. The following three sections contain the numerical techniques to find this solution and
to overcome a few potentially destructive numerical difficulties. In Sect. 4 we apply Gaussian
integration to compute a class of integrals that reduce the problem to a set of non–linear equa-
tions,andinSect. 5wedescribeaconvenientapproachforthesolutionoftheseequations. Next,
in Sect. 6, we show how to tailor the root finding routine to the hierarchical structure of IFS
attractors. Asaresult,weobtaintheequilibriummeasureontheselatter. Picturesandgraphsof
numerical experiments illustrate the theory. In section 7 we compute the potential generated in
the complexplanebytheequilibriummeasureonthe IFSattractorandwecomputeits capacity.
The application of this approach to deep questions on equilibrium measures on Cantor sets is
briefly discussed in the conclusions.
2 Attractors of Iterated Function Systems
Let us therefore construct the set E of which we want to place an electrostatic charge. To do
this, we need to recall, as briefly as possible and in the simplest setting, the construction of
Iterated Function Systems (IFS) [26, 19, 14, 12, 13]. These are collections of maps φ :R R,
i
→
i = 1,...,M, for which there exists a set , called the attractor of the IFS, that solves the
A
equation
= φ ( ):=Φ( ). (3)
i
A A A
[
j=1,...,M
When the maps φ are contractive, the attractor is unique. It can also be seen as the fixed
i
pointoftheoperatorΦ,definedineq. (3),ontheseAtofcompactsubsetsofR. Sincethisspaceis
complete in the Hausdorff metric, and since Φ turns out to be contractive in this metric, the set
can be also found as the limit of the sequence Φn(E0), where E0 is any non-empty compact
A
set:
= lim Φn(E0). (4)
A n→∞
2
Attractors of Iterated Function Systems feature a rich variety of topological structures, so
that their full characterization is far from being fully understood, especially in the case of IFS
with uncountably many maps [21, 22, 17, 18]. We nonetheless restrict ourselves in this paper to
a particular family, that of IFS with a finite number of affine maps of the form:
φ (s)=δ (s γ )+γ , j =1,...,M, (5)
j j j j
−
where δ are real numbers between zero and one, called contraction ratios, and γ are real con-
i i
stants, that geometrically correspond to the fixed points of the maps. Under these conditions,
the attractor is a finite or infinite collection of intervals, or a Cantor set. We will consider
A
this last,interestingcase. It canbe easilyrecognizedthat the famous “middle–thirdCantorset”
follows in this class: it can be obtained by taking just two maps with δ = 1/3 and γ = 1,
1
−
γ =1. We shall study this case later in this paper.
2
As remarkedabove,anynon-empty compactset E0 canbe used in eq. (4). We letE0 be the
convex hull of the attractor , that can be easily be identified as the interval
A
E0 =[γ ,γ ], (6)
1 M
where we have ordered the IFS maps according to increasing values of their fixed points: γ <
j
γ , for any j = 1,...,M 1. This is the core of the common hierarchical construction of the
j+1
−
set , as the limit of the sequence of compact sets En:
A
Mn
En =Φn(E0)= [α ,β ]. (7)
i i
[
i=1
In the case of fully disconnected IFS (i.e. those for which the intervals φ (E0) are pairwise
j
disjoint), En is the union of N := Mn disjoint intervals, that we will denote as En := [α ,β ].
i i i
In other words, the set En can be seen as a vector of intervals, stored via the two vectors of
their extreme points. Of course, these latter should also carry a superscript n, referring to
the generation. Not to overburden the notation, we will leave this superscript implicit, when
confusion is not possible. We restrict ourselves to the disconnected IFS case, for the remainder
of the paper.
For similarity with the spectral analysis of periodic solids, we will call the intervals En at
i
r.h.s. of eq. (7) bands at generation (or level) n. It appears from eq. (7) that all bands En
i
at level n can be obtained by applying the transformations φ , j = 1,...,M, to the bands at
j
generation n 1. As a consequence, the length of these intervals, as well as their sum, tend to
−
zero geometrically and the attractor is a Cantor set of null Lebesgue measure.
WhilenoneofthebandsEn−1arealsobandsatleveln,thecontraryhappensfortheso–called
i
gaps between the bands. For these, we will use the notation:
Mn−1
Gn := Gn = (β ,α ). (8)
i i i+1
[ [
i i=1
At level zero we have no bands, while at level one these are in the number of M 1. It is
−
then noticeable that all gaps at level n 1 remain gaps at level n, and at the same time a set
−
of new gaps is generatedfrom the former: let Hn be the set of “new” gaps createdat level n, so
that
Gn =Gn−1 Hn. (9)
[
The latter set, Hn, is iteratively constructed as
M
Hn = φ (Hn−1), (10)
j
[
j=1
starting from H1 =(β1,α1) ... (β1 ,α1 ). This seemingly complicated algorithm is noth-
1 2 M−1 M
ing more than the straightfoSrwardSgeneralization of the construction of the ternary Cantor set
by deleting the “middlethird”. This propertyofthe setofgapsis importantinourcomputation
of the equilibrium measure on the Cantor set .
A
3
3 Equilibrium measure on a set of intervals
Let us now embed En in the complex plane, and study its equilibrium measure. The solution
of the equilibrium problem for a finite union of N intervals [α ,β ], i = 1,...,N, is well known
i i
[3, 5, 24, 25, 4]. Define the polynomial Y(z),
N
Y(z)= (z α )(z β ), (11)
i i
− −
Y
i=1
anditssquareroot, Y(z),astheonewhichtakesrealvaluesforzrealandlarge(largerthanγ
M
indeed). Also, let thpe real number ζ belong to the open interval (β ,α ), for i=1,...,N 1
i i i+1
−
(i.e. to the gapGn). Define Z(z)as the monic polynomialof degree N 1 with roots atall ζ ’s:
i − i
N−1
Z(z)= (z ζ ). (12)
i
−
Y
i=1
With these premises, there exists a unique set of values ζ ,i=1,...,N 1 that solve the set
i
{ − }
of coupled, non–linear equations
ai+1 Z(s)
ds=0, i=1,...,N 1. (13)
Zbi |Y(s)| −
p
Our first task in the following will be to evaluate numerically these integrals for any given set of
values ζ ,i=1,...,N 1 . The secondtaskwillbe tofindthe solutionofthis setofequations.
i
{ − }
Thisisofparamountimportance,foritpermitsustofindtheequilibriummeasureofthesetEn:
for simplicity of notation, we denote it by σn :=σEn:
N
1 Z(s)
dσn(s)= χ (s) | | ds. (14)
π X [αi,βi] Y(s)
i=1 | |
p
It is apparent from the previous equation that the measure σn is absolutely continuous with
respect to the Lebesque measureon En. Of particularrelevance are alsothe integralsof σn over
the intervals composing En, that we call the harmonic frequencies:
1 βi Z(s)
ωn := | | ds. (15)
i π Zαi |Y(s)|
p
In fact, when these frequencies are rational numbers, of the kind p /N, with p integer, there
i i
existsastrict-TpolynomialonEn[4],thatis,apolynomialwithoscillationpropertiesmimicking,
and extending, those of the classical Chebyshev polynomials on a single interval. Also, we will
consider the integrated measures Ωn:
m
m
Ωn := ωn, (16)
m i
X
i=1
that are the integral of σn on the interval [γ ,β ].
1 m
Finally, the Green function with pole at infinity can be computed as the complex integral
z Z(s)
G(z)= ds, (17)
Zα1 Y(s)
p
in whichno absolutevalue appears. TheGreenfunctioncanbe usedto compute the logarithmic
capacity of the set En, C(En), via a real integral
α1 Z(s) 1
log(C(En))= [ ]ds. (18)
Z−∞ Y(s) − s−(α1+1)
p
Althoughourtechniquecouldpermittoevaluatetheaboveintegral,wewillcomputethecapacity
C(En) following a different approach.
4
4 Integrating the ratio Z/√Y
The first step in the determination of the equilibrium measure is the numericaldetermination of
integralsofthekind(13),orofthekind(15),thatarequitesimilar. Astraightforwardtechnique
to achieve this goal is described in this section. The first integrals are:
1 Z(s)
:= ds, (19)
Ki π ZGi |Y(s)|
p
where G is a gap at generation n. The case when G is replaced by a band E can be handled
i i i
by minor variations of the technique that we are going to describe. For simplicity of notation,
since n is fixed, we will omit its mention in the following derivation. A full set of integrals ,
i
K
for variable i,needto be evaluated. Let us now keepthe index i fixedandshow howto compute
any single one of them.
Firstly, consider the change of variables induced by the affine transformation ψ := x
i
→
A x+B , that maps the interval G into E0 = [γ ,γ ]. Without any loss of generality we can
i i i 1 M
assume that E0 = [ 1,1]. The coefficients A and B can be computed by careful bookkeeping
i i
−
from the map parameters δ and β . Even if we feel no need to report the explicit formulae
j j
here, they are essential in the numerical implementation that we will describe in the following.
In so doing, the bands E and the gaps G , for m different than i, are mapped, via ψ , to
m m i
intervals outside [ 1,1]. Equivalently, the variables ζ are mapped via the same map to new
m
values ψ (ζ ). New−functions Z¯ and Y¯ are then written as in eqs. (11) and (12), now in terms
i m i i
of the transformed ψ (ζ ), ψ (α ) and ψ (β ). In conclusion, the integral can be written as
i k i m i m i
K
1 1 Z¯ (s)
i
= ds. (20)
Ki π Z−1 Y¯i(s)
| |
p
Secondly, remark that the product 1 z2 appears in the rescaled function Y¯(z). Therefore,
i
we can part the full product Y¯(z) in two−factors:
i
Y¯(z)=(1 z2)Y˜(z), (21)
i i
−
where Y˜(z) is implicitly defined. With these notations, it appears clearly that the integration
i
of Z¯i(s) with respect to the Lebesgue measure on [ 1,1] can be seen as an integration with
√|Y¯i(s)| −
respect to the Chebyshev measure ds/π√1 s2. Then, the integral can be approximated by
i
− K
a Gaussian summation,
1 Z¯ (s) ds K Z¯ (x )
i i k
= w , (22)
Ki Z−1 Y˜i(s) π√1−s2 ≃kX=1 k Y˜i(xk)
q| | q| |
where x ,w are Gaussian points and weights for the Chebyshev measure, respectively. As a
k k
consequence, each integral can be readily evaluated by Gaussian summation.
i
K
Clearly, the above assumes that the values ζ be known for all i=1,...,N 1. To compute
i
−
these latter, the N equations = 0, i = 1,...,N 1, must be solved. In the next section
i
K −
we employ an algorithm for the solution of this set of non-linear equation that requires the
computation of the derivatives ∂ /∂ζ . Also these quantities can be computed via Gaussian
i m
K
summation. Observe in fact that the partial derivative of eq. (22) requires the computation of
′
∂
Z¯(z)= A (z ψ (ζ )), (23)
i i i l
∂ζ − −
m Y
l6=m
that therefore leads to
∂Ki A K w ′l6=m(xk−ψi(ζl)). (24)
∂ζm ≃− iXk=1 kQ Y˜(xk)
q| |
5
5 Solving the non-linear equations = 0
i
K
We have seen that the determination of the equilibrium measure on the set En requires the
solution of a system of N 1 non–linear functions in N 1 variables, (ζ ,...,ζ ) = 0.
i 1 N−1
We can compute this soluti−on by a careful usage of the rou−tine HYBRJKin Minpack [2], which
employs a modification of the Powell hybrid method. This technique is a variation of Newton’s
method, that iteratively converges to the solution vector. As it turns out, HYBRJ cannot
be applied blindly without taking into account the particular nature of the set En and of the
discretized integrals (22). In fact, in the construction of the attractor of a fully disconnected
IFS, all bands En have a length that goes to zero geometrically fast, while gaps differ in length
i
by orders of magnitude. This requires a series of procedures, in the evaluation of the integrals
and ∂Ki, that we explain in this section.
Ki ∂ζm
Thefirstconsistsintheapproachdescribedattheendoftheprecedingsection: itisconvenient
to scale the each interval Gn to [ 1,1], when performing the relative integration, to employ the
i −
Gaussian form in eqs. (22) and (24).
When this is done, a further difficulty is to be overcome: the function Z¯ (x )/ Y˜(x ) is
i k i k
q| |
made of factors differing by various orders of magnitude, that must be conveniently rearranged
when evaluating it numerically. Therefore, in the expansion
Z¯ (x ) N−1(x ψ (ζ ))
i k = l=1 k− i l (25)
Q
q|Y˜i(xk)| q|xk−ψi(αi)||xk−ψi(βm+1)| m6=i,i+1|xk−ψi(αm)||xk−ψi(βm)|
Q
we find it convenient to group together the ratios (x ψ (ζ ))/ x ψ (α ) x ψ (β ),
k i m k i m k i m
− | − || − |
for m<i, and (x ψ (ζ ))/ x ψ (α ) x ψ (β ), fpor m>i, in which numerator
k i m k i m+1 k i m+1
− | − || − |
and denominator have comparpable size. The more the index m is different from i, the larger are
the differences between x andthe points ψ (α )and ψ (β ), andthe morethese ratiostend to
k i m i m
one. Physically, this means that remote gaps Gn have a small influence on the integral : our
m Ki
ordering correctly reproduces this fact. As it appears from eq. (25), grouping terms as above
leaves out a few factors, those close to the interval ψ (Gn) = [ 1,1], that mostly contribute to
i i −
the value of the full product, and are therefore taken care separately. A similar procedure can
also be applied to the formula for derivatives, eq. (24).
Butthemoreimportantprecautiontobetakenfollowsfromconsideringthevariablesinvolved
in the Newton-like iteration of HYBRJ. In the original equations =0 the unknowns are the
i
K
positionsoftheN 1zerosofZ,i.e. ζ ,m=1,...,N 1. EachofthesebelongstothegapGn.
− m − m
It is then apparent, from the description in Sect. 2, that the range of variation of these values
can differ by orders of magnitude, and can quickly become smaller than numerical precision. It
isthenmandatorytoresorttonewvariables: weconsiderinplaceofζ thenormalizedvariables
m
λ := ψ (ζ ), where, as above, ψ is the affine transformation that maps Gn into [ 1,1].
m m m m m −
These new variables are all bound to the interval [ 1,1]. Some care must now be taken in
−
the evaluation of the compounded derivatives, but fortunately, the new variables are the most
natural, in the construction of the IFS, as it can be easily recognized from eqs. (5), (7).
The above provides us with a working code for the computation of the non-linear functions
(22) and of the partial derivatives (23), as functions of the variables λ , m=1,...,N 1, that
m
canbelinkedasasubroutinetothefortran(yes,fortran!) codeHYBRJ,runindouble−precision
on a 32-bit processor. In Figure (1) we show the absolute values of the integrals before and
i
after the call to HYBRJ in a typical case, that of the ternary Cantor set at generKation number
n = 7, to be defined more precisely later on in Sect. 7. The algorithm has converged to the
correct solution within numerical precision.
To fully appreciate the precision of the solution vector, we can examine the value of the
partialderivatives ∂Ki atthe solutionvector. ThesevaluesarereportedinFigure(2)versusthe
∂λm
difference i m: as remarked above, the larger this difference (in absolute value) the lesser the
−
influence ofλ onthe integral : this isclearlyevidentinthe figure. Also,wecanobservethat
m i
K
the system of equations is almost “diagonal”,in the sense that diagonalcomponents with i=m
(the tipofthe “Christmastree”)areordersofmagnitudelargerthanthe non-diagonalbranches,
i=m.
6
6
0.01
0.0001
1e-006
1e-008
εi 1e-010
1e-012
1e-014
1e-016
1e-018
0 20 40 60 80 100 120 140
i
Figure1: Absolutevalueǫ ofthenon-linearfunctions attheinitialvector(λ ,...,λ )(see
i i 1 N−1
K
next section for its definition) (green crosses), and after the action of the root finding routine
(red crosses). The case under study is the ternary Cantor set at generation number n=7. The
number of Gaussian points employed is 2048.
1
0.1
0.01
0.001
0.0001
Ki,m 1e-005
1e-006
1e-007
1e-008
1e-009
1e-010
-150 -100 -50 0 50 100 150
i-m
Figure 2: Absolute value K of the partial derivatives ∂Ki versus i m at the final solution
i,m ∂λm −
vector, for the same case of Fig. 1.
7
0.09
0.08
0.07
0.06
0.05
nm 0.04
λ
0.03
0.02
0.01
0
-0.01
0 1 2 3 4 5 6 7 8 9
n
Figure 3: Roots λn of the equations (13) versus generation number n, for m =1,...,2n−1 1,
m −
fortheIFSdescribedinthetext. Linesconnectrootsinthesamegapatdifferentn. Thenumber
of Gaussian points employed is 160.
6 Hierarchical structure of the equilibrium measure
We have describedin the previous sectionan algorithmfor the determination of the roots of the
equilibriumequations(13). The algorithmapplies to anyfinite family ofintervals. Inparticular,
wewanttoapplyittoasystemofintervalsgeneratedbyafullydisconnectedIFS.Intheprevious
section we have already presented results in Figures 1 and 2 referring to the case of the middle–
third Cantorset. Afurther remarkonthe waythey have been obtainedis important. While the
case of the ternary Cantor set is particularly favorable because of its symmetry, we now choose
to describe our procedure in the case of an asymmetric IFS given by two-maps with δ = 4/5
1
and δ =1/10. We will return to the middle–third Cantor set in the next section.
2
ObservethatthecardinalityofEn isMn,whereM isthenumberofIFSmaps,andtherefore
thenumberofgapsisMn 1. Becauseofeq. (9),asubsetofcardinalityMn−1 1oftheselatter
− −
exist already as gaps at generation n 1. The old gaps are approximately (M 1) times less
− −
numerous than the new ones,nonetheless, their rˆole is crucial. In fact, the Newton-like searchof
the algorithm HYBRJ in Minpack need to be initialized. It is particularly efficient to initialize
the values of the roots λn to the “old” values at generation λn−1, and the new ones to zero
Mm m
(which means that ζ lies in the middle of Gn.)
m m
EverythingdescribedisclearlyevidentinFig. 3,thatplotsthevaluesofthesolutionvariables
λn versus n. We can observe the new gaps that are created at each new generation: data are
m
drawn as crosses, and lines join the λ values of roots of Z lying in the same gap for successive
values of n. As a gap gets “old”, the root inside it tends to a limit value. From Fig. 3 we can
also observe that all roots have absolute value less than 1/10, while in principle they are bound
by one.
After all this preparatoryworkwe are now ready to examine the equilibrium measure on the
set En. It is straightforwardto obtain the harmonicfrequencies ωn andthe integratedmeasures
i
Ωni, discussed in the second section, via the Gaussian integration technique described in Sect.
m
4. Thecaseoffig. 3isexaminedagaininfigure4,byplottingtheintegratedmeasuresΩn versus
m
n: recall that Ωn is the measure of the set (γ ,ζn) under the equilibrium measure on En. As
m 1 m
8
1.1
1
0.9
0.8
0.7
m
n
Ω
0.6
0.5
0.4
0.3
0.2
0 1 2 3 4 5 6 7 8 9
n
Figure 4: Integrated measures Ωn versus versus generation number n, for the IFS described in
m
the text. Lines connect points as explained in the text, and as done in the previous figure 3.
They are only apparently horizontal (see the following Fig. 6). The number of Gaussian points
employed is 160.
such, these values are the heights of the plateaus in Fig. 5, that draws the integral
x
Ωn(x)= dσn(s), (26)
Z
γ1
versus x, for the same IFS described above.
As n grows, En tends in Hausdorff distance to the attractor of the IFS, that is a Cantor set
of zero Lebesgue measure. We want now to show that our technique can be used to compute
the equilibrium measure ofthe IFS attractoritself: it appearsfrom Fig. 5 that Ωn(x) tends to a
limit function Ω(x) forLebesgue a.e. x. This functionis preciselythe integralofthe equilibrium
measure on the IFS attractor.
Infact,wehavealreadyremarkedthatgaps,oncecreated,stayforeverinthe complementary
oftheCantorset. Letusthereforegaugetheconvergence,asngrows,oftheequilibriummeasure
of the infinite intervalto the left ofeachgap. This is nothing else thanΩn ,when the index of
m(n)
the gap, m, depends on the generation n according to the rule m(n+1)=Mm(n). In Figure 6
we choose the first gap, G1, that is labeled at successive generations as Gn : we extract a single
1 2n
line from fig. 5 and we replot it alone so to render evident its variation. It appears that we have
exponential convergence of the equilibrium measures, to the limit equilibrium measure of the
Cantor set. We have found the same behavior for all gaps, so that we can numerically observe
convergence to the limit function Ω(x) . But measures are born for integration, and integrating
is what we are now up to.
7 The Capacity of En and of the IFS attractor
The convergence that we have observed at the end of the previous section is just an instance
of the convergence of σn to the equilibrium measure on the IFS attractor. A second example
is obtained considering the electrostatic potential V(σn;z) defined in eq. (1). Our technique
permits to compute it in the full complex plane. Care must be exerted when z belongs to En,
9
1
0.9
0.8
0.7
0.6
n(x) 0.5
Ω
0.4
0.3
0.2 n=1
n=2
n=3
0.1 n=4
n=5
n=6
0
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
x
Figure 5: Harmonic measures Ωn(x) versus x at various generation numbers n for the IFS
described in the text. The number of Gaussian points employed is 160.
0.753
0.752
0.751
0.75
m
n
Ω
0.749
0.748
0.747
0.746
1 2 3 4 5 6 7
n
Figure 6: Integrated measure Ωn versus generation number n for the IFS described in the text
2n
(red crosses). Also plotted is the fit of the discrete values by the curve f(n)=a+be−cn (green
line). The number of Gaussian points employed is 160.
10