Connectivity of the Julia set for the Chebyshev-Halley family on degree n polynomials

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