Connec i i y o he Julia se o he Chebyshe -Halley
amily on deg ee npolynomials
∗B. Campos, †J. Canela and ∗P. Vindel
camp[email p o ec ed], [email p o ec ed], [email p o ec ed]
Abs ac
We s udy he Chebyshe -Halley amily o oo inding algo i hms om he poin o
iew o holomo phic dynamics. Nume ical expe imen s show ha he speed o con e gence
o he oo s may be slowe when he basins o a ac ion a e no simply connec ed. In
his pape we p o ide a c i e ion which gua an ees he simple connec i i y o he basins o
a ac ion o he oo s. We use he c i e ion o he Chebyshe -Halley me hods applied o
he deg ee npolynomials zn+c, ob aining a cha ac e iza ion o he pa ame e s o which
all Fa ou componen s a e simply connec ed and, he e o e, he Julia se is connec ed. We
also s udy how inc easing na ec s he dynamics.
1 In oduc ion
Mos o he p oblems aced by scien is s and enginee s in ol e equa ions ha do no ha e a
known analy ical solu ion. Nume ical me hods a e a good op ion o ackle and sol e eal wo ld
p oblems. In pa icula , i e a i e me hods a e used o ind app oxima ions o he solu ions o
(z) = 0.
The bes -known oo - inding algo i hm is New on’s me hod, which has o de o con e gence
2. Many nume ical me hods o o de h ee o mo e a e de i ed om New on’s scheme: Cheby-
she me hod, also known as supe -New on me hod (see [15], o example), Halley’s me hod and
supe -Halley me hod. A mo e de ailed s udy o he cons uc ion and e olu ion o hese nume -
ical me hods can be seen in [13]. These me hods belong o a amily o nume ical algo i hms
called he Chebyshe -Halley amily, which is gi en by
xn+1 =xn−1 + 1
2
L (xn)
1−αL (xn) (xn)
0(xn),(1)
whe e
L (xn) = (xn) 00 (xn)
( 0(xn))2
∗Ins i u o de Ma em´a icas y Aplicaciones de Cas ell´on (IMAC), Uni e si a Jaume I. Spain
†Uni e si ´e Pa is-Es Ma ne-la-Vall´ee. F ance
1
and α∈C. Wi hin his amily, Chebyshe me hod is ob ained o α= 0, Halley’s me hod is
ob ained o α=1
2and supe -Halley me hod is ob ained o α= 1. Mo eo e , as α ends o ∞
hese algo i hms con e ge o New on’s me hod.
Failu es in in e media e calcula ions made by a compu e a e e y di icul o de ec . Con-
sequen ly, one o he common aims o nume ical analysis is o selec obus algo i hms, ha is,
algo i hms wi h a good nume ical s abili y in a wide ange o si ua ions. The e o e, i is usual
o conside i e a i e me hods wi h high o de o con e gence. Howe e , he adii o con e -
gence which ensu e ha he solu ion o he me hod is co ec may dec ease when we inc ease
he o de o he me hod. One way o add ess his issue is o s udy he nume ical me hod om
a dynamical poin o iew, i.e., o conside he i e a i e me hod as a disc e e dynamical sys em
and o s udy i s s abili y. This is a line o wo k ha has p o en o be especially ui ul in
ecen yea s (see, o example, he pape s [1], [2], [3], [6], [7], [9], [14]).
In his pape we s udy he Chebyshe -Halley amily om a dynamical poin o iew in o de
o ind which membe s o he amily ha e be e s abili y. To ca y ou his dynamical s udy,
he oo - inding algo i hm is applied o a polynomial P. By doing so, we ob ain a a ional map
Q:b
C→b
C, whe e b
Cdeno es de Riemann sphe e, whose dynamics desc ibes how beha es he
me hod when applied o he polynomial P. Indeed, he poin s which con e ge o he oo s o
Pwhen applying o he nume ical me hod a e exac ly hose which con e ge o he oo s o P
when i e a ing he map Q.
We shall gi e a sho in oduc ion o he concep s used in holomo phic dynamics. A mo e
de ailed desc ip ion o he opic can be ound in [4] and [17]. We conside he disc e e dynamical
sys em gi en by he i e a es o a a ional map Q:b
C→b
C. A poin z0∈b
Cis called ixed i
Q(z0) = z0and pe iodic o pe iod p > 1 i Qp(z0) = z0and Qk(z0)6=z0 o k < p. In he la e
case, we say ha hz0i={z0, Q(z0),··· , Qp−1(z0)}is a p-cycle. A poin is p epe iodic i i is
e en ually mapped unde i e a ion o Qon o a pe iodic poin . The mul iplie o a ixed poin
z0is gi en by λ=Q0(z0). Analogously, he mul iplie o a p-cycle hz0iis gi en by (Qp)0(z0).
We say ha a ixed poin o a cycle is a ac ing i |λ|<1 (supe a ac ing i λ= 0), epelling
i |λ|>1, and indi e en i |λ|= 1. In he la e case λ=e2πiθ, whe e θ∈[0,1). We say
ha an indi e en poin o cycle is a ionally indi e en o pa abolic i θ∈Q. Any a ac ing
o pa abolic poin z0has a basin o a ac ion, an open se o poin s which con e ge unde
i e a ion o Q o z0, ela ed o i . We deno e i by
A(z0) = {z∈b
C:Qk(z)→z0when k→∞}.
The basin o a ac ion o an a ac ing (o pa abolic) p-cycle can be de ined analogously using
ha all elemen s o hz0ia e a ac ing (o pa abolic) ixed poin s o Qp.
The dynamics o a ional map Qspli s he Riemann sphe e in o wo o ally in a ian sub-
se s. The Fa ou se ,F(Q), consis s o he z∈b
Csuch ha he amily o i e a es o Q,
{Q(z), Q2(z), . . . , Qk(z), . . .}, is no mal, o equi alen ly equicon inuous, in some open neigh-
bou hood Uo z. The Fa ou se is open and co esponds o he se o poin s wi h s able
dynamics. I s complemen , he Julia se J(Q), is closed and co esponds o he se o poin s
which p esen chao ic beha iou . The connec ed componen s o he Fa ou se , called Fa ou
componen s, a e mapped among hemsel es unde i e a ion. All Fa ou componen s o a a-
ional map a e ei he pe iodic o p epe iodic ([19]). By he Classi ica ion Theo em (see e.g.
[17]), all pe iodic Fa ou componen s a e ei he basins o a ac ion o a ac ing o pa abolic
cycles, o simply connec ed o a ion domains (Siegel disks) o doubly connec ed o a ion do-
mains (He man ings). Mo eo e , all pe iodic Fa ou componen s a e ela ed o a c i ical poin ,
ha is, a poin z∈b
Csuch ha Q0(z) = 0. Indeed, he basin o a ac ion o an a ac ing o
2
a pa abolic cycle con ains, a leas , a c i ical poin . Also, he o bi o , a leas , a c i ical poin
accumula es on he bounda y o a Siegel disk o a He man ing.
When a oo - inding algo i hm is s udied om he poin o iew o holomo phic dynamics,
i is usual o apply he me hod o low deg ee polynomials (see o ins ance [11], [6], [7], [9]).
The eason is ha when he deg ee o he polynomial inc eases, he numbe o c i ical poin s
o he a ional map ob ained also inc eases. This is a se ious d awback when analysing he
pa ame e spaces o he me hods applied o high deg ee polynomials, since he o bi s o he
c i ical poin s a e c ucial o es ablish he exis ence o basins o a ac ion which do no come
om he oo s.
In he pape [5], we conside he Chebyshe -Halley amily o nume ical me hods applied o
he deg ee npolynomials zn+c, whe e z∈Cand c∈C {0}. As a as we know, his is he
i s ime ha a s udy o a amily o oo - inding algo i hms applied o deg ee npolynomials
is conside ed om he poin o iew o dynamical sys ems and he co esponding pa ame e
spaces a e p o ided. In Figu e 1 we show he pa ame e spaces o his amily o di e en alues
o n. Despi e o he inc ease in he numbe o c i ical poin s, we use he symme ies o he
dynamical sys em ob ained o jus i y ha i is enough o ollow he o bi o a single c i ical
poin o de e mine he exis ence o basins o a ac ion o he han he ones p o ided by he
oo s. This p ope y allows us o ob ain pic u es o he pa ame e spaces o polynomials o his
amily. We use hese nume ical s udies oge he wi h some heo e ical esul s o p o ide a i s
analysis o how he se s o pa ame e s wi h good dynamical beha iou a y as ninc eases. Fo
ixed n, we show ha he e exis s a unique pa ame e α o which he Chebyshe -Halley me hod
has o de o con e gence 4. Howe e , he dynamics o such poin s may a y when we inc ease
n. Fo ins ance, o α= 1 (supe -Halley me hod) he algo i hm has o de o con e gence 4 o
n= 2, bu i p esen s bad dynamical beha iou o big n(see Figu es 9, 10 and 11).
The goal o his pape is o con inue he esea ch began in [5]. We ocus on he s udy
o pa hological dynamical beha iou which appea s bo h in he dynamical and he pa ame e
planes. Fi s , we s udy he dynamical condi ions which lead o non-simply connec ed basins o
a ac ion o he oo s. I has been nume ically obse ed ha he basins o a ac ion may ha e
holes, which seem o lead o slowe speed o con e gence. In Sec ion 3 we p o ide a dynamical
condi ion (P oposi ion 3.1) which can be applied o any holomo phic amily o a ional maps
wi h a single ee c i ical poin (modulo symme ies). Using his condi ion, we p o e ha he
basins o a ac ions o he oo s o he Chebyshe -Halley amily a e mul iply connec ed i
and only i he immedia e basin o a ac ion o he oo z= 1 con ains ano he c i ical poin
c6= 1, and no p eimage o z= 1 o he han himsel . This cha ac e iza ion is used o s udy he
connec i i y o hei Julia se s. In Theo em 3.9 we p o e ha he Julia se o he Chebyshe -
Halley ope a o is disconnec ed i and only i he p e ious condi ion holds. These esul s allow
us o loca e he alues o he pa ame e o which he Julia se is disconnec ed, and he e o e,
he nume ical me hods a e mo e uns able.
A e wa ds, we s udy he pa ame e s which p esen bad beha iou when d awing he pa-
ame e planes. Pa ame e planes o he Chebyshe -Halley amily applied o zn−1 a e shown
in Figu e 1, o se e al alues o n. The d awings a e ob ained as ollows. Fo each pa ame e
αwe i e a e a c i ical poin up o 150 imes. I he o bi o he c i ical poin con e ges o a
oo ( ha is, he i e a e wsa is ies |w−ξ|<10−4 o some n h- oo o he uni y ξ) in less
han 150 i e a es, we plo he co esponding poin using a scale om ed ( as con e gence) o
g een o pu ple and o g ey (slow con e gence). I a e 150 i e a es he c i ical o bi has no
con e ged o a oo , we plo he poin in black. See Sec ion 5.1 o a mo e de ailed explana ion
on how he images o pa ame e planes a e p oduced. We obse e in ed pa ame e s o which
3
−1 0 1 2 3 4
−2
−1
0
1
2
(a) n=2
−1 0 1 2 3 4
−2
−1
0
1
2
(b) n=3
−1 0 1 2 3 4
−2
−1
0
1
2
(c) n=5
−1 0 1 2 3 4
−2
−1
0
1
2
(d) n=10
−1 0 1 2 3 4
−2
−1
0
1
2
(e) n=25
−1 0 1 2 3 4
−2
−1
0
1
2
( ) n=100
Figu e 1: Pa ame e spaces o he Chebyshe -Halley amily applied o zn−1.
4
he co esponding c i ical poin s con e ge as o he oo s. We can also obse e he Ca se ,
he se o pa ame e s o which he c i ical poin s do no belong o he basins o a ac ion o
he oo s. In all he igu es we can dis inguish wo big disks called he head an he body o
he Ca . These se s co espond o pa ame e s o which s ange ixed poin s a e a ac ing (see
P oposi ion 2.3 and P oposi ion 2.4). The Ca se consis s o he head and he body (wi h hei
deco a ions) and a necklace-like s uc u e which su ounds he head, ha we call he Colla .
Fo nsmall, we obse e he Colla colou ed in yellow. Howe e , o n∈ {25,100} he e appea
some black disks which do no co espond o s able beha iou , as is he case o he head and
he body. In his pape we analyse a ound which bi u ca ion pa ame e s hese egions appea
(see Sec ion 4 and 5.1), and s udy he associa ed ope a o s om a nume ical poin o iew
(Sec ion 5.2).
The pape is s uc u ed as ollows. In Sec ion 2 we ecall he main p ope ies o he ope a o s
ob ained applying he Chebyshe -Halley me hods o he polynomials zn+cand analyse he
s abili y o he s ange ixed poin s. In Sec ion 3 we p o ide a dynamical condi ion o he
simple connec i i y o he basins o a ac ion o he oo s and he connec i i y o he Julia
se . In Sec ion 4 we s udy di e en bi u ca ion pa ame e s wi hin he Colla o he Ca se .
Finally, in Sec ion 5 we s udy he pa ame e and he dynamical planes o he amily om a
nume ical poin o iew.
2 Dynamical s udy on deg ee npolynomials
We s udy is he Chebyshe -Halley amily o nume ical me hods applied on he polynomials
zn+c. Resul s o his amily applied on polynomials o deg ee wo can be seen in [11], [10],
[12].
In [5] we show ha he ope a o ob ained o (z) = zn−1 is conjuga e o he one ob ained
when applying he Chebyshe -Halley me hod o (z) = zn+c,c∈C−{0}. The e o e, i is
enough o s udy he Chebyshe -Halley me hods o zn−1 o unde s and hem o all zn+c,
whe e c∈C−{0}.
We deno e by On,α(z) he ixed poin ope a o ob ained o zn−1.By subs i u ing (z) =
zn−1 in (1) we ha e:
On,α(z) = z−(zn−1)((−1+2α+n−2αn) + (1 −2α−3n+ 2αn)zn)
2nzn−1(α(n−1)(zn−1) −nzn)=
=(1 −2α)(n−1) + (2 −4α−4n+ 6αn −2αn2)zn+ (n−1)(1 −2α−2n+ 2αn)z2n
2nzn−1(α(1 −n)+(−α−n+αn)zn).(2)
Excep o degene a e cases s udied in Sec ion 4, he deg ee o his ope a o is 2n. The e o e,
i has 2n+ 1 ixed poin s and 4n−2 c i ical poin s.
The ixed poin s a e ob ained by sol ing On,α(z) = z. On one hand, we ob ain he n h - oo s
o he uni y, co esponding o he ze os o he polynomial zn−1, which a e supe a ac ing
ixed poin s. The o he n+ 1 ixed poin s a e z=∞and he nsolu ions o he equa ion
−1+2α+n−2αn + (1 −2α−3n+ 2αn)zn= 0.(3)
These poin s a e called s ange ixed poin s, because hey do no ma ch wi h he solu ions o
he polynomial. The c i ical poin s o he ope a o On,α a e he solu ions o O0
n,α(z) = 0, whe e
O0
n,α(z) = (zn−1)2(n−1)(α(1 −2α)(n−1)2+ (1 −2n−2α+ 2αn) (−α−n+αn)zn)
2nzn(α(n−1) + (α+n−αn)zn)2.(4)
5

The n h- oo s o he uni y a e double c i ical poin s and, hence, a e supe a ac ing ixed poin s
o local deg ee 3. The poin z= 0 is a c i ical poin o mul iplici y n−2 since i is mapped
wi h deg ee n−1 o z=∞. This asse ion ollows om he e m 1/zn−1on Equa ion (2). The
emaining nc i ical poin s a e gi en by
cn,α,ξ =cξ=ξα(n−1)2(2α−1)
n(2n−1) −α(4n−1)(n−1) + 2α2(n−1)21/n
,(5)
whe e ξdeno es an n h- oo o he uni y, ı.e. ξn= 1. The exis ence o any s able beha iou o
he dynamical sys em o he han he basins o a ac ion o he ze os o zn−1 is con olled by
he o bi s o hese n ee c i ical poin s.
The o de o con e gence o he oo s o all membe s he Chebyshe -Halley amily is a
leas 3. The nex esul , which co esponds o [5, P oposi ion 6.3], s a es ha he e is a single
pa ame e o which he o de o con e gence inc eases o 4. In his case, he n h- oo s o he
uni y a e c i ical poin s o mul iplici y h ee and a e supe a ac ing ixed poin s o local deg ee
4.
P oposi ion 2.1. Fo n≥2, he ope a o On,α has o de o con e gence 4 i and only i
α=2n−1
3n−3.
The ollowing lemma, which co esponds o [5, Lemma 6.2], s a es ha he dynamics o he
maps On,α is symme ic wi h espec o he n h- oo o he uni y. I ollows om he lemma
ha he n ee c i ical o bi s a e symme ic and, he e o e, i is enough o con ol one o hem.
Lemma 2.2. Le n∈Nand le ξbe an n h- oo o he uni y, i.e. ξn= 1. Then Iξ(z) = ξz
conjuga es On,α(z)wi h i sel , i.e.
Iξ◦On,α(z) = On,α ◦Iξ(z).
The s abili y o he ixed poin z=∞was s udied in [5], whe e we p o ed he ollowing
p oposi ion.
P oposi ion 2.3. The ixed poin z=∞sa is ies he ollowing s a emen s.
1. I α−1−4n+5n2
2(n−1)(2n−1)<n
2(2n−1), hen z=∞is an a ac o . In pa icula , i is a supe a -
ac ing ixed poin i i s mul iplie is equal o 0, i.e. α=n
n−1.
2. I α−1−4n+5n2
2(n−1)(2n−1)=n
2(2n−1), hen z=∞is an indi e en ixed poin .
3. I α−1−4n+5n2
2(n−1)(2n−1)>n
2(2n−1), hen z=∞is a epelling ixed poin .
We inish his sec ion s udying he s abili y o he o he s ange ixed poin s, which a e
gi en by he solu ions o (3).
P oposi ion 2.4. The ns ange ixed poin s a e gi en by
z∗
ξ=ξ1−2α−n+ 2αn
1−2α−3n+ 2αn1/n
,(6)
whe e ξdeno es an n h- oo o he uni y, and sa is y he ollowing s a emen s.
6
1. I α−2n−1
n−1<1
2, hen hey a e a ac ing ixed poin s. In pa icula , hey a e supe a -
ac ing ixed poin s i α=2n−1
n−1.
2. I α−2n−1
n−1=1
2, hen hey a e indi e en ixed poin s.
3. I α−2n−1
n−1>1
2, hen hey a e epelling ixed poin s.
P oo . The de i a i e o he ope a o is gi en in Equa ion (4). Subs i u ing zn=1−2α−n+2αn
(1−2α−3n+2αn)
in (4), we ha e ha
O0
n,α(z∗
ξ) = −2−2α(n−1) + 4n
n−1.
W i ing α=a+ib and de eloping he equa ion O0
n,α(z∗
ξ)= 1 we ob ain
a−2n−1
n−12
+b2=1
4,
which is he equa ion o he ci cle cen ed a 2n−1
n−1wi h adius 1/2. The s ange ixed poin s a e
a ac ing inside his ci cle, since O0
n,α(z∗
ξ)<1.They a e supe a ac ing i O0
n,α(z∗
ξ) = 0, which
is sa is ied o α=2n−1
n−1. Ou side his ci cle hey sa is y O0
n,α(z∗)>1, so hey a e epelling.
Finally, on he ci cle hey sa is y O0
n,α(z∗
ξ)= 1, so hey a e indi e en ixed poin s.
P oposi ion 2.3 explains he e olu ion o he loca ion and size o he head o he Ca se .
In pa icula , i shows ha he adius o he co esponding disk dec eases o 1
4as n→ ∞. On
he o he hand, in P oposi ion 2.4 we show ha he body o he Ca se is a disk o adius 1
2,
which does no depend on n.
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(a) n= 3
0 0.5 1 1.5 2
−1
−0.5
0
0.5
1
(b) n= 10
Figu e 2: Dynamical planes o On,α o α=2n−1
n−1.
In Figu e 2 we show he dynamical planes o On,α o n∈ {3,10}and α=2n−1
n−1, which
co espond o he supe a ac ing case in P oposi ion 2.4. We obse e wi h a scaling om ed
7
o g een o pu ple and o g ey he poin s which con e ge o he he oo s in 75 i e a es, ha
is, he i e a e wsa is ies |w−ξ|<10−4 o some n h- oo o he uni y ξ. We obse e in black
he poin s which ha e no con e ged o he oo s a e 75 i e a es. The basins o a ac ion o
he s ange ixed poin s appea as black inge s coming om in ini y. No ice ha he s ange
ixed poin s, and he co esponding basins o a ac ion, a e symme ically loca ed. We e e
o Sec ion 5.2 o a mo e de ailed explana ion on how he images a e p oduced.
3 Simple connec i i y o Fa ou componen s
Nume ical expe imen s show ha he speed o con e gence o he oo s may be slowe when
he basins o a ac ion a e no simply connec ed, ha is, he e a e connec ed componen s o
he basins o a ac ion which ha e holes. In his sec ion we gi e a dynamical condi ion which
cha ac e izes whe he he immedia e basins o a ac ion o he ope a o s On,α a e simply
connec ed. We also p o e ha he ope a o s On,α can no ha e He man ings, ha is, doubly
connec ed o a ion domains. The exis ence o such domains would lead o a posi i e measu e
se o ini ial condi ions no con e ging o he oo s. Fo comple eness, we end his sec ion
showing how he p e ious esul s lead o a dynamical cha ac e iza ion o he connec i i y o
he co esponding Julia se s.
3.1 Simple connec i i y o immedia e basins o a ac ion o supe -
a ac ing ixed poin s
In his subsec ion we gi e a cha ac e iza ion o he simple connec i i y o he immedia e basin
o a ac ion o a suppe a ac ing ixed poin in he case ha i can con ain a mos one c i ical
poin besides he suppe a ac ing ixed poin i sel . This esul is s a ed o a gene al a ional
map . Fo simplici y, we assume ha i has a suppe a ac ing ixed poin a z= 0. We
deno e by A (0) i s basin o a ac ion and by A∗
(0) i s immedia e basin o a ac ion, ha is,
he connec ed componen o A (0) which con ains z= 0.
P oposi ion 3.1. Le :b
C→b
Cbe a a ional map and le z= 0 be a supe a ac ing ixed
poin o . Assume ha A (0) con ains a mos one c i ical poin o he han z= 0. Then,
exac ly one o he ollowing s a emen s holds.
1. The se A∗
(0) con ains no c i ical poin o he han z= 0. Then, A∗
(0) is simply con-
nec ed.
2. The se A∗
(0) con ains a c i ical poin c6= 0 and a p eimage z0o z= 0,z06= 0. Then,
A∗
(0) is simply connec ed.
3. The se A∗
(0) con ains a c i ical poin c6= 0 and no p eimage o z= 0 o he han z= 0
i sel . Then, A∗
(0) is mul iply connec ed.
The p oo o P oposi ion 3.1 uses he Riemann-Hu wi z o mula (see [4]). Gi en a p ope
map :U→V, his o mula ela es he connec i i ies o Uand V. The connec i i y o an
open se U⊂b
Cis gi en by he numbe o connec ed componen s o ∂U. A map is p ope i
he p eimages o compac se s a e compac se s. Gi en a a ional map , an open se Vand a
connec ed componen Uo −1(V), he map :U→Vis p ope .
8
Theo em 3.2 (Riemann-Hu wi z Fo mula).Le Uand Vbe wo connec ed domains o b
Co
ini e connec i i y mUand mVand le :U→Vbe a deg ee kp ope map b anched o e c
c i ical poin s coun ed wi h mul iplici y. Then
mU−2 = k(mV−2) + c.
P oo o P oposi ion 3.1. We conside he B¨o che coo dina e o he supe a ac ing ixed
poin z= 0 (see [17, Theo em 9.1]). The B¨o che coo dina e is a con o mal map φ:U→D ,
whe e Uis a simply connec ed neighbou hood o z= 0, 0 < ≤1, and D is he open disk
o cen e 0 and adius . This map conjuga es wi h z→zd, whe e dis he local deg ee
o he supea a ac ing ixed poin z= 0. Mo e speci ically, o all zwi h |z|< we ha e
φ◦ ◦φ−(z) = zn. In pa icula , φ(0) = 0.
We now assume ha Uis he maximal domain o de ini ion o he B¨o che coo dina e
and, hence, is maximal. I A∗
(0) does no con ain any ex a c i ical poin , hen = 1 and
U=A∗
(0) (see [17, Theo em 9.3]). In pa icula , A∗
(0) is simply connec ed. On he o he
hand, i A∗
(0) con ains an ex a c i ical poin c6= 0, hen < 1, U⊆A∗
(0), and c∈∂U (see
[17, Theo em 9.3]). Le V= (U). Then, γ0:= ∂V is a simple closed cu e (i is he p eimage
unde φo he ci cle o adius d). Deno e by γ1 he connec ed componen o −1(γ0) which
con ains ∂U. Then, γ1is he union o wo simple closed cu es which in e sec a he c i ical
poin c, say γ1
1, and γ2
1. Fu he mo e, ei he ∂U =γ1
1o ∂U =γ1
1∪γ2
1=γ1(see Figu e 3).
0γ0
γ2
1
γ1
1
z0
γ2
1
γ1
1
0γ0
U
VV
U
Figu e 3: The wo possibili ies o he con igu a ion o γ1=γ1
1∪γ2
1.
Assume ha ∂U =γ1
1and deno e by U0 he connec ed componen o b
C γ2
1which does no
con ain z= 0. Since (γ1
1) = (γ2
1) = γ0, i ollows ha (U0) = (U) = V. In pa icula U0
con ains a p eimage z0o 0. Mo eo e , U∪U0⊂A∗
(0) and we can ake a simply connec ed
neighbou hood U ⊂ A∗
(0) o U∪U0. Deno e by W he connec ed componen o −1(U) ha
con ains 0. Since does no ha e any c i ical poin in A∗
(0) o he han cand 0, i ollows
om he Riemann-Hu wi z o mula (Theo em 3.2) ha |W:W → U has deg ee d+ 1 and W
is simply connec ed. No ice ha he deg ee o |Wis a leas d+ 1 since z= 0 is a p eimage
o mul iplici y do i sel and Wcon ains an ex a p eimage z0o 0. Repea ing he p ocess,
we ob ain a sequence o simply connec ed domains U=U0⊂U1⊂ ··· ⊂ Un⊂ ···, whe e
Unis a connec ed componen o −1(Un−1) and Un−1⊂Un. Mo eo e , A∗
(0) = S∞
n=0 Un, om
which we conclude ha A∗
(0) is a simply connec ed se con aining a p eimage z0o z= 0 and
a c i ical poin c6= 0.
Now assume ha ∂U =γ1
1∪γ2
1=γ1. Then, he connec ed componen s W1and W2o b
C γ1
1
and b
C γ2
1, espec i ely, which do no con ain z= 0 a e mapped unde on o open se s which
con ain b
C V. In pa icula , bo h W1and W2con ain pa o he Julia se J( ) and, he e o e,
9
(1 −n)α(1 −2n+ 2α(n−1))(n−1)
n+α(1 −n)−(2n−1 + α(n−1) (n−2))2
=
=nq(α(n−1) −1)2+ 2 (n−1)2
,
which implies
(1 + 2α(n−1) −2n) (n−1)2n2= 0.
The oo o he las equa ion is
α=2n−1
2n−2.
Howe e , his is no a alid solu ion since we a e conside ing he case α6=2n−1
2n−2.
Finally, we conside he alue α=n
n−1disca ded in Equa ion (8). In his case, he deg ee
o he ope a o emains 2n:
On, n
n−1(z) = (n+ 1) + 2(n2−1)zn−(n−1) z2n
2n2zn−1.
Then, he only alues o which he ope a o degene a es o a a ional unc ion o lowe
deg ee a e α=1
2and α=2n−1
2n−2.
In he ollowing we analyse he dynamical beha iou o he nume ical me hods co esponding
o he bi u ca ion pa ame e s ob ained abo e.
Lemma 4.2. Fo α=1
2 he s ange ixed poin s a e z= 0 and z=∞. Mo eo e , he only
c i ical poin s a e he oo s o he polynomial, which ha e mul iplici y 2
P oo . The a ional unc ion o α=1
2is gi en by
On, 1
2(z) = (n+ 1)z+ (n−1)zn+1
(n−1) + (n+ 1)zn.(9)
The ixed poin s o On, 1
2(z) a e he oo s, z= 0 ( he ns ange ixed poin s collide a 0) and
z=∞. The de i a i e is gi en by
O0
n, 1
2
(z) = (n2−1) (zn−1)2
((n−1) + (n+ 1)zn)2.(10)
The only c i ical poin s, which a e he ze os o O0
n, 1
2
(z), a e he oo s o he polynomial zn−1
and ha e mul iplici y 2.
Lemma 4.3. Fo α=2n−1
2n−2, he oo s o he polynomial a e c i ical poin s o mul iplici y 2.
Mo eo e , he poin s {0,∞} o m a cycle o pe iod 2, which is supe a ac ing i n > 2.
P oo . The ope a o o α=2n−1
2n−2is
On, 2n−1
2n−2(z) = 1 + (2n−1)zn
(2n−1 + zn)zn−1.(11)
16

The ixed poin s a e he oo s o he polynomial and he n h- oo s o −1. The de i a i e is
gi en by
O0
n, 2n−1
2n−2
(z) = −(2n2−3n+ 1) (zn−1)2
zn(2n−1 + zn)2.(12)
The oo s o he polynomial a e c i ical poin s wi h mul iplici y 2. Since he nume a o o
On, 2n−1
2n−2(z) does no anish a z= 0 and z= 0 is a oo o mul iplici y n−1 o he denomina o ,
we conclude ha z= 0 is mapped o z=∞wi h deg ee n−1. The e o e, z= 0 is a c i ical
poin i n−1>1, ha is, n > 2. Simila ly, z=∞is mapped o z= 0 wi h deg ee n−1 and
is a c i ical poin i n > 2. In pa icula {0,∞} is a pe iod wo cycle. Since a pe iodic cycle
is supe a ac ing when any o he poin s o he cycle is c i ical, we conclude ha {0,∞} is a
supe a ac ing cycle i n > 2.
No ice ha he poin z=∞is a ixed poin o all α6=2n−1
2n−2bu i becomes a pe iod 2
supe a ac ing pe iodic poin i n > 2 and α=2n−1
2n−2(see Lemma 4.3). This may lead o e y
uns able dynamics (compa e Figu e 7).
We now look o bi u ca ion pa ame e s o which he ee c i ical poin s coincide wi h z= 0
o a e mapped on o i . No ice ha z= 0 is a p eimage o he ixed poin z=∞. In ha case,
he ee c i ical poin s a e p epe iodic and, he e o e, he e is no o he s able beha iou han
he basins o a ac ion o he oo s.
Lemma 4.4. The ee c i ical poin s collapse wi h z= 0 i and only i α= 0.
P oo . In he exp ession o he c i ical poin s (Equa ion (5)) we obse e ha c i ical poin s
become equal o 0 o α= 0 and α=1
2. Ne e heless, α=1
2is a degene a e case; in his case,
as we ha e seen in Lemma 4.2, z= 0 is no a c i ical poin . So, he only alue o which ee
c i ical poin s collapse wi h z= 0 is α= 0.
Now, we calcula e he alues o he pa ame e such ha c i ical poin s a e p eimages o
z= 0. As be o e, o such pa ame e s he c i ical poin s a e p epe iodic since z= 0 maps o
he ixed poin z=∞.
Lemma 4.5. C i ical poin s a e p eimages o z= 0 i and only i α=1±√2(1−n)
n−1.
P oo . Fo α6={n
n−1,2n−1
2n−2}, he ee c i ical poin s a e gi en by
cn,α,ξ =ξ (n−1)2(−1+2α)α
n(2n−1) −(4n−1) (n−1) α+ 2 (n−1)2α2!1/n
,(13)
whe e ξdeno es an n h- oo o he uni y. We disca d he pa ame e α=n
n−1since he c i ical
poin s coincide wi h z=∞(see P oposi ion 2.3). We also disca d he pa ame e 2n−1
2n−2since he
deg ee o On,α dec eases and he e a e no ee c i ical poin s (see Lemma 4.1 and Lemma 4.3).
P eimages o z= 0 a e ob ained om On,α (z) = 0, ha is
(1 −2α)(n−1) + (2 −4α−4n+ 6αn −2αn2)zn+ (n−1)(1 −2α−2n+ 2αn)z2n= 0.
The solu ions sa is y
zn=−1+2n+α(n−1) (n−2) ±nq2 (n−1) + (1 + α(1 −n))2
(n−1) (1 −2n+ 2α(n−1)) .
17
The solu ions coincide wi h c i ical poin s i
−1+2n+α(n−1) (n−2) ±nq2 (n−1) + (1 + α(1 −n))2
(n−1) (1 −2n+ 2α(n−1)) =
=(n−1)2(2α−1) α
n(2n−1) −(4n−1) (n−1) α+ 2 (n−1)2α2.
Ope a ing and simpli ying we ob ain
(2α−1) (1 + 2α(n−1) −2n)2(n−1)2α2(n−1)2−2α(n−1) + 2n−1= 0.
The solu ions o he las equa ion a e
α=1
2, α =2n−1
2 (n−1), α =1±p2 (1 −n)
n−1.
The pa ame e s α=1
2and α=2n−1
2n−2a e degene a ed cases wi h no ee c i ical poin s (see
Lemma 4.1).
We conclude ha c i ical poin s a e he p eimages o z= 0 i and only i α=1±√2(1−n)
n−1.
No ice ha he alues o αin oduced in Lemma 4.5 coincide, in he pa ame e space, wi h
wo symme ic poin s o he Colla o he Ca se , whe e he main ami ica ions appea (see
Figu e 6).
5 Nume ical s udies and conclusions
The goal o his sec ion is o pe o m a nume ical s udy on he Chebyshe -Halley amily.
We i s analyse he pa ame e planes nea he bi u ca ion pa ame e s desc ibed in Sec ion 4.
A e wa ds we s udy he dynamical planes o he amily. We ocus on he dynamics o maps
wi h a disconnec ed Julia se and he e olu ion when ng ows o he dynamics o some ele an
membe s o he Chebyshe -Halley amily.
5.1 Pa ame e planes
The d awings o pa ame e planes o he ope a o s On,α along he pape a e done using a
p og am w i en in C which wo ks as ollows. We ake a g id o poin s (1500 ×1000 poin s
o Figu e 1, 1200 ×2000 poin s o Figu e 6, and 1500 ×1500 poin s o Figu e 7). Then,
we associa e a pa ame e α∈C o each poin o he g id. The ange o he eal and he
imagina y pa o he pa ame e s αis indica ed in he ho izon al and e ical axes o he
images, espec i ely. Fo ixed αwe compu e one o he n ee c i ical poin s and i e a e i up
o 150 imes. A each i e a ion we e i y i he i e a ed poin whas con e ged o any o he
n h- oo s o he uni y (we e i y i |w−ξ|<10−4, o any n h- oo o he uni y ξ). I he
c i ical o bi con e ges o a oo o he uni y, we colou he co esponding poin wi h an scaling
om ed ( as con e gence) o yellow, g een, blue, pu ple and o g ey (slow con e gence). I
a e 150 i e a ions he o bi o he c i ical poin has no con e ged o a oo , we plo he poin
in black. In Figu e 1 we show he pa ame e planes o he ope a o On,α o di e en alues o
n.
18
−0.5 0 0.5
−1
−0.5
0
0.5
1
(a) n= 10
−0.5 0 0.5
−1
−0.5
0
0.5
1
(b) n= 25
−0.5 0 0.5
−1
−0.5
0
0.5
1
(c) n= 100
Figu e 6: Zooms in on he pa ame e planes nea he main bi u ca ion poin s o he Colla .
The aim o his subsec ion is o analyse he pa ame e planes o he amily nea he bi-
u ca ion pa ame e s ound in Sec ion 4. In Figu e 6 we show zooms in on he pa ame e
planes o se e al n. The anges o αinclude he bi u ca ion pa ame e s α= 0 (Lemma 4.4),
α= 1/2 (Lemma 4.1) and α±=1±√2(1−n)
n−1(Lemma 4.5). The pa ame e α= 1/2 (Halley’s
me hod) appea s in he igu es as a ip o an an enna which joins i wi h α= 0 (Chebyshe ’s
me hod). We obse e ha he bi u ca ion s uc u e a α= 1/2 is a he simple, which could
indica e s abili y o he amily o pa ame e s nea Halley’s alue. On he o he hand, a mo e
complex bi u ca ion s uc u e appea s a Chebyshe ’s pa ame e (α= 0). Recall ha o his
pa ame e all c i ical poin s collide a z= 0, which is a p eimage o he ixed poin z=∞.
Indeed, o n= 25 we obse e a black disk o pa ame e s o which he o bi o he c i ical
poin has no con e ged o a oo a e 150 i e a es. This black disk does no co espond o
any s able beha iou , bu o pa ame e s o which he c i ical o bi would equi e mo e i e a es
o con e ge o a oo . Indeed, his disk o pa ame e s dec eases when we inc ease he numbe
o i e a ions. A simila si ua ion occu s nea he pa ame e s α±, which a e ma ked wi h whi e
poin s in Figu e 6. These pa ame e s co espond o ope a o s o which he c i ical poin s a e
p eimages o z= 0. Se e al o hese black egions no co esponding o s able beha iou appea
when d awing he pa ame e planes o big n, being he bigge ones a ound α±. Nume ical
expe imen s seem o indica e ha he bigges black egions no ela ed o s able beha iou ap-
pea a ound pa ame e s o which he ee c i ical poin s a e e en ually mapped unde i e a ion
o he co esponding ope a o on o z= 0. In Sec ion 5.2 we show he dynamics o On,α o all
he bi u ca ion pa ame e s discussed in his pa ag aph in o de o analyse i hey ha e o be
a oided o i hey p esen good dynamical beha iou .
In Figu e 7 we show zooms in on he pa ame e plane nea he pa ame e α=2n−1
2n−2. I
n > 2, his bi u ca ion pa ame e co esponds o a lowe deg ee ope a o o which {0,∞} is
a supe a ac ing cycle. The e o e, his is a pa ame e o be a oided. In Figu e 7 we indica e
i wi h a small whi e poin . We obse e how, o n= 10 and n= 25, his pa ame e is he
19
1 1.02 1.04 1.06 1.08 1.1
−4
−2
0
2
4
·10−2
(a) n= 10
0.96 0.98 1 1.02 1.04
−4
−2
0
2
4
·10−2
(b) n= 25
Figu e 7: Zooms in on he pa ame e planes nea he degene acy pa ame e α=2n−1
2n−2.
cen e o a cascade o bi u ca ions. Simila ly o wha happens in he pa ame e plane nea he
o he bi u ca ion pa ame e s, his cascade o bi u ca ions may lead o a black egion which does
no co espond o s able beha iou (see Figu e 7 (b)). This cascade o bi u ca ions akes place
in a a he small egion which is adjacen o he hype bolic componen o pa ame e s which
con ains α=2n−1
3n−3.Fo ixed n, his pa ame e co esponds o he single ope a o On,α wi h
o de o con e gence 4 o he oo s (see P oposi ion 2.1). Mo eo e , he pa ame e s wi hin his
hype bolic componen ha e connec ed Julia se (see Co olla y 3.11), which a p io i makes o
hem desi able pa ame e s. Howe e , he hype bolic componen seems o become smalle and
o mo e sligh ly o he le when we inc ease n(see Figu e 1). In pa icula , he pa ame e
α= 1, which has o de o con e gence 4 o he oo s o n= 2, alls in o he cascade o
bi u ca ions o n= 25 (see Figu e 7). In Sec ion 5.2 we show how he dynamics o α= 1
e ol es when we inc ease nand i app oaches and en e s he cascade o bi u ca ions.
5.2 Dynamical planes
In his sec ion we p o ide nume ical d awings o dynamical planes o he ope a o s On,α. The
d awings o dynamical planes a e done using a p og am w i en in C which wo ks as ollows.
We ake a g id o 1500 ×1500 and we associa e a poin z∈C o each poin o he g id. The
ange o he eal and he imagina y pa o he poin s zis indica ed in he ho izon al and
e ical axes o he images, espec i ely. Then, we i e a e he poin zup o 75 imes. A each
i e a ion we e i y i he i e a ed poin whas con e ged o any o he n h- oo s o he uni y
(we e i y i |w−ξ|<10−4, o any n h- oo o he uni y ξ). I i con e ges o any oo o he
uni y, we colou he co esponding poin in he g id wi h an scaling om ed ( as con e gence)
o yellow, g een, blue, pu ple and o g ey (slow con e gence). I a e 75 i e a ions he o bi
has no con e ged o a oo , we plo he poin in black.
The goal o his sec ion is o analyse he dynamical beha iou o he ope a o s On,α o se -
e al pa ame e s in o de o ha e a be e idea o which pa ame e s p o ide a good dynamical
beha iou . We i s in es iga e nume ically o which pa ame e s he Julia se is disconnec ed
20
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(a) n= 3, α = 0.2+1.4i
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(b) n= 3, α = 2i
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(c) n= 25, α = 0.2+1.4i
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(d) n= 25, α = 2i
Figu e 8: Dynamical planes o se e al On,α(z) wi h a disconnec ed Julia se .
and we analyse hei dynamics. Be o e doing so, we ecall he concep o hype bolic componen
o pa ame e s. A hype bolic componen is a connec ed se o pa ame e s o which all c i ical
o bi s accumula e on a ac ing o supe a ac ing cycles. The Julia se is s able wi hin hype -
bolic componen s, i.e. he Julia se is connec ed o one pa ame e o he componen i and only
i i is connec ed o all pa ame e s o he componen . Hype bolic componen s o pa ame e s
o which he ee c i ical poin s belong o he basins o a ac ion o he oo s appea in ed in
Figu e 1. In Co olla y 3.11 we ha e p o en ha he ope a o s co esponding o he hype bolic
componen which con ains α=2n−1
3n−3ha e connec ed Julia se . Nume ical simula ions seem
o indica e ha he o he ed hype bolic componen s bounded by he Colla o he Ca se
co espond o pa ame e s o which he ee c i ical poin s belong o he basins o a ac ion o
he oo s bu no o hei immedia e basins o a ac ion and, hence, co espond o ope a o s
wi h a connec ed Julia se (see Theo em 3.9). The only hype bolic componen co esponding
o ope a o s wi h disconnec ed Julia se seems o be he ed unbounded componen on he
complemen o he Colla (see Figu e 1). In Figu e 8, Figu e 9 ( ), and Figu e 11 ( ) we show
21

he dynamical planes o se e al pa ame e s wi hin his hype bolic componen o n= 3 and
n= 25. We ake he pa ame e α= 0.2 + 1.4i, which is close o he Colla , and he pa ame e s
α= 2iand α= 4i, which a e a he away om he Colla . Fo α= 0.2 + 1.4i he holes in
he basins o a ac ion a e ela i ely big and a e easy o obse e. Fo he pa ame e s α= 2i
and α= 4i hese holes a e much mo e di icul o see since hey become smalle . Howe e , he
con e gence o he oo s wi hin his las pa ame e s is slowe . Indeed, o n= 3 we obse e
how he basins o a ac ions o he oo s ha e a mo e in ense ed onali y o α= 0.2 + 1.4i,
which indica es as con e gence. Fo n= 25 we can obse e how he basins o a ac ion o
he oo s acqui e a blueish onali y o α= 2iand, pa icula ly, o α= 4i, which indica es
slowe con e gence o he oo s. We may conclude ha , wi hin his hype bolic componen ,
pa ame e s close o he Colla ha e bigge holes on he basins o a ac ion o he oo s, bu
he con e gence o he oo s is as e .
In Figu e 9, Figu e 10 and Figu e 11 we show he dynamical planes o he ope a o On,α o
se e al o he mos ele an pa ame e s s udied along he pape o n= 3, n= 10 and n= 25,
espec i ely. The pa ame e s analysed a e he ollowing. Sub igu e (a) co esponds o α=2n−1
3n−3,
which p o ides he me hod wi h o de o con e gence 4 o he oo s (see P oposi ion 2.1).
Sub igu es (b), (c), and (d) co espond o α= 1/2 (Halley’s me hod), α= 0 (Chebyshe ’s
me hod) and α= 1 (Supe -Halley me hod), espec i ely. Sub igu e (e) co esponds o α=
1+√2(1−n)
n−1, which is a bi u ca ion pa ame e o which he ee c i ical poin s a e mapped o
z= 0 (see Lemma 4.5). Finally, sub igu e ( ) co esponds o α= 4iwhich is a pa ame e in he
unbounded componen o he complemen o he Colla o which he Julia se is disconnec ed.
A p io i, he pa ame e ha should p o ide a be e dynamical beha iou is α=2n−1
3n−3
(sub igu e (a)), since he co esponding ope a o s ha e o de o con e gence 4 o he oo s.
We obse e ha , nea he oo s, i is he pa ame e wi h a as es a e o con e gence (i has
he mos in ense ed nea he oo s). Howe e , he global dynamics a e complex. Indeed, o
n= 10 and n= 25 we obse e how he Julia se becomes complica ed and he e appea black
disks o ini ial condi ions which do no con e ge o he oo s a e 75 i e a es. We wan o
poin ou ha i we inc ease he numbe o i e a es we can make disappea his disk o n= 10.
Ne e heless, o n= 25 he disk dec eases e y slowly when we inc ease he numbe o i e a es.
This is a kind o pa hological beha iou which is no ela ed o s able beha iou di e en om
he oo s and can lead o big se s o ini ial condi ions which do no con e ge o he oo s.
The pa ame e s α= 0 (Chebyshe ’s me hod, sub igu e (c)), α=1+√2(1−n)
n−1(sub igu e (e))
and α= 4i(sub igu e ( )) p esen a simila dynamical beha iou . F om hem, he one which
p esen s a la ge black disk a ound z= 0 is α= 0. This indica es ha Chebyshe ’s me hod
may no be an ou s anding oo inding algo i hm o apply o he amily zn+cwhen nis
big. On he o he hand, he pa ame e α=1+√2(1−n)
n−1, which is a bi u ca ion pa ame e in he
Colla , p esen s a be e dynamical beha iou han he pa ame e α= 4i, which is a pa ame e
co esponding o an ope a o wi h a disconnec ed Julia se . Indeed, he speed o con e gence
o he oo s o he ope a o s co esponding o Chebyshe ’s me hod and α=1+√2(1−n)
n−1is much
as e han he one o α= 4i. We can conclude ha he black egions which appea a ound
bi u ca ion pa ame e s in he pa ame e plane (see Figu e 1 and Figu e 6) a e no necessa ily
ela ed o pa icula ly bad dynamical beha iou and hey a e p e e able han he pa ame e s
co esponding o disconnec ed Julia se .
The pa ame e α= 1 is pa icula ly in e es ing. Fo n= 2 he co esponding ope a o
has o de o con e gence 4 o he oo s. I s ill p esen s e y good dynamical beha iou o
22
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(a) α= 5/6
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(b) α= 0.5, Halley’s me hod
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(c) α= 0, Chebyshe ’s me hod
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(d) α= 1, Supe -Halley me hod
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(e) α= (1 + i√4)/2
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
( ) α= 4i
Figu e 9: Dynamical planes o On,α(z) o n= 3 and di e en alues o α.
23
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(a) α= 19/27
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(b) α= 0.5, Halley’s me hod
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(c) α= 0, Chebyshe ’s me hod
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(d) α= 1, Supe -Halley me hod
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(e) α= (1 + i√18)/9
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
( ) α= 4i
Figu e 10: Dynamical planes o On,α(z) o n= 10 and di e en alues o α.
24
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(a) α= 49/72
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(b) α= 0.5, Halley’s me hod
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(c) α= 0, Chebyshe ’s me hod
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(d) α= 1, Supe -Halley me hod
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
(e) α= (1 + i√48)/24
−3−2−1 0 1 2 3
−3
−2
−1
0
1
2
3
( ) α= 4i
Figu e 11: Dynamical planes o On,α(z) o n= 25 and di e en alues o α.
25