centers

Centers of components of Mandelbrot set

Center of n-period hyperbolic component of Mandelbrot set is a point of parameter plane :

which is mapped by multiplier map on the center of unit circle m=0+0*i (in multiplier plane) c: m(c,F,period) = 0
for which periodic orbit is superattractive = multiplier is equal to zero c: l_n(c) = 0

We will use second definition to compute centers.
because :

l_n(c) = F'_M(n,c,Z0)
F'_M(Z0,c,n):= 2ⁿ*Z0*Z1*.. *Z_n-1;

so multiplier of periodic orbit is equal to product of first derivative of Mandelbrot function at periodic points
l_n(c) = 2ⁿ*Z0*Z1*.. *Z_n-1

So l_n(c) = 0
only if z=0 belongs to periodic orbit
it means that z=0 is periodic point of Mandelbrot function:
F[n, c, z=0] = 0

/* basic funtion = monic and centered complex quadratic polynomial
http://en.wikipedia.org/wiki/Complex_quadratic_polynomial */
f(z,c):=z*z+c $
/* iterated function */
fn(n, z, c) :=
if n=1 then f(z,c)
else f(fn(n-1, z, c),c) $
/* roots of Fn are periodic point of fn function */
Fn(n,z,c):=fn(n, z, c)-z $

/* G = irreducible divisors of Fn */

polynomial G :
- roots of G are periodic points of period p ( exactly, not for p and its divisors)
- it's order ( degree ) is slightly lower then degree of Fn

for i:1 thru 15 step 1 do display(divisors(i))$
divisors(1)={1}
divisors(2)={1,2}
divisors(3)={1,3}
divisors(4)={1,2,4}
divisors(5)={1,5}
divisors(6)={1,2,3,6}
divisors(7)={1,7}
divisors(8)={1,2,4,8}
divisors(9)={1,3,9}
divisors(10)={1,2,5,10}
divisors(11)={1,11}
divisors(12)={1,2,3,4,6,12}
divisors(13)={1,13}
divisors(14)={1,2,7,14}
divisors(15)={1,3,5,15}

Fn(1) = c = G(1)
Fn(2) = c*(c+1) = G(1)* G(2)
Fn(3) = c*(c^3+2*c^2+c+1) = G(1)* G(3)
Fn(4) = c*(c+1)*(c^6+3*c^5+3*c^4+3*c^3+2*c^2+1) = G(1)* G(2)* G(4)
Fn(5) = c*(c^15+8*c^14+28*c^13+60*c^12+94*c^11+116*c^10+114*c^9+94*c^8+69*c^7+44*c^6+26*c^5+14*c^4+5*c^3+2*c^2+c+1) = = G(1)* G(5)
Fn(6) = c*(c+1)*(c^3+2*c^2+c+1)*(c^27+13*c^26+78*c^25+293*c^24+792*c^23+1672*c^22+2892*c^21+4219*c^20+5313*c^19+5892*c^18+5843*c^17+5258*c^16+ 4346*c^15+3310*c^14+2331*c^13+1525*c^12+927*c^11+536*c^10+298*c^9+155*c^8+76*c^7+35*c^6+17*c^5+7*c^4+3*c^3+c^2-c+1)= G(1)* G(2)* G(3) * G(6)

Fn(7) = c*(c^63+32*c^62+496*c^61+4976*c^60+36440*c^59+208336*c^58+971272*c^57+3807704*c^56+12843980*c^55+37945904*c^54+99582920*c^53+ 234813592*c^52+502196500*c^51+981900168*c^50+1766948340*c^49+2943492972*c^48+4562339774*c^47+6609143792*c^46+8984070856*c^45+ 11500901864*c^44+13910043524*c^43+15941684776*c^42+17357937708*c^41+17999433372*c^40+17813777994*c^39+16859410792*c^38+ 15286065700*c^37+13299362332*c^36+11120136162*c^35+8948546308*c^34+6939692682*c^33+5193067630*c^32+3754272037*c^31+2625062128* c^30+1777171560*c^29+1166067016*c^28+742179284*c^27+458591432*c^26+275276716*c^25+160617860*c^24+91143114*c^23+50323496*c^22+ 27049196*c^21+14162220*c^20+7228014*c^19+3598964*c^18+1749654*c^17+831014*c^16+385741*c^15+175048*c^14+77684*c^13+33708*c^12+ 14290*c^11+5916*c^10+2398*c^9+950*c^8+365*c^7+132*c^6+42*c^5+14*c^4+5*c^3+2*c^2+c+1) = Gn(1)* Gn(7)

Fn(8) = c*(c+1)*(c^6+3*c^5+3*c^4+3*c^3+2*c^2+1)*(c^120+60*c^119+1770*c^118+34250*c^117+489375*c^116+5511102*c^115+50989975*c^114+ 398959395*c^113+2696755065*c^112+16009488760*c^111+84574656816*c^110+401877006480*c^109+1733107463040*c^108+6834914821260*c^107 +24811070182620*c^106+83371586399039*c^105+260615957165336*c^104+761188082391138*c^103+2085341983243658*c^102+ 5377303587067725*c^101+13092095906249710*c^100+30181158127871065*c^99+66047545613729349*c^98+137526242897521387*c^97+ 273055685652049512*c^96+517973316808768672*c^95+940463418168922896*c^94+1637134234590816416*c^93+2736605515930668712*c^92+ 4399030584560979688*c^91+6809393331711977377*c^90+10162926651591927936*c^89+14642200658326619088*c^88+ 20387237220949711442*c^87+27462099883710267563*c^86+35823271041713949254*c^85+45296178536637687081*c^84+ 55565997236718038571*c^83+66187135731800060711*c^82+76612820738925961694*c^81+86242488958463274826*c^80+ 94481112789965725168*c^79+100801976018285941440*c^78+104803422336957136740*c^77+106250977507865123996*c^76+ 105098793143557019895*c^75+101487983924396807096*c^74+95723317903066081340*c^73+88233057204728849436*c^72+ 79518917115832434169*c^71+70103821471231936142*c^70+60484437480620582867*c^69+51093714978195649483*c^68+ 42276335962975311453*c^67+34277623335577859252*c^66+27244489096635136284*c^65+21235681116184560728*c^64+ 16237990133298034432*c^63+12185127163725489840*c^62+8976499782758573072*c^61+6493890970072701905*c^60+4614878920815010176* c^59+3222581751710173440*c^58+2211880343460599616*c^57+1492636547692947392*c^56+990597359091017086*c^55+646699117035044471* c^54+415409562944557103*c^53+262617111655159022*c^52+163433639370729534*c^51+100144730020422499*c^50+60433299357506429*c^49+ 35923338854117473*c^48+21038627397895636*c^47+12141832138153068*c^46+6906542871283655*c^45+3872827682093776*c^44+ 2141241178405040*c^43+1167479848735120*c^42+627846072983732*c^41+333079003836572*c^40+174340431656069*c^39+90046721164189* c^38+45900127501242*c^37+23093412618276*c^36+11469308277609*c^35+5623447798913*c^34+2722207351629*c^33+1301145233716*c^32+ 614110737048*c^31+286228647553*c^30+131751286720*c^29+59896973608*c^28+26896860416*c^27+11931393376*c^26+5229116872*c^25+ 2264514156*c^24+969160031*c^23+409965164*c^22+171422294*c^21+70854302*c^20+28948921*c^19+11691330*c^18+4667641*c^17+1842415*c^16 +718975*c^15+277264*c^14+105560*c^13+39636*c^12+14680*c^11+5368*c^10+1944*c^9+698*c^8+248*c^7+84*c^6+28*c^5+8*c^4+1)

G(1) = Fn(1) = c
G(2) = Fn(2)/G(1) = c+1
G(3) = Fn(3)/G(1) = c^3+2*c^2+c+1
G(4) = Fn(4)/(G(1)*G(2)) = c^6+3*c^5+3*c^4+3*c^3+2*c^2+1
G(5) = Fn(5)/G(1) = c^15+8*c^14+28*c^13+60*c^12+94*c^11+116*c^10+114*c^9+94*c^8+69*c^7+44*c^6+26*c^5+14*c^4+5*c^3+2*c^2+c+1
Gn(6) = Fn(6)/(G(1)*G(2)*G(3)) = c^27+13*c^26+78*c^25+293*c^24+792*c^23+1672*c^22+2892*c^21+4219*c^20+5313*c^19+5892*c^18+5843*c^17+5258*c^16+ 4346*c^15+3310*c^14+2331*c^13+1525*c^12+927*c^11+536*c^10+298*c^9+155*c^8+76*c^7+35*c^6+17*c^5+7*c^4+3*c^3+c^2-c+1
G(7) = Fn(7)/G(1) = c^63+32*c^62+496*c^61+4976*c^60+36440*c^59+208336*c^58+971272*c^57+3807704*c^56+12843980*c^55+37945904*c^54+99582920*c^53+ 234813592*c^52+502196500*c^51+981900168*c^50+1766948340*c^49+2943492972*c^48+4562339774*c^47+6609143792*c^46+8984070856*c^45+ 11500901864*c^44+13910043524*c^43+15941684776*c^42+17357937708*c^41+17999433372*c^40+17813777994*c^39+16859410792*c^38+ 15286065700*c^37+13299362332*c^36+11120136162*c^35+8948546308*c^34+6939692682*c^33+5193067630*c^32+3754272037*c^31+2625062128* c^30+1777171560*c^29+1166067016*c^28+742179284*c^27+458591432*c^26+275276716*c^25+160617860*c^24+91143114*c^23+50323496*c^22+ 27049196*c^21+14162220*c^20+7228014*c^19+3598964*c^18+1749654*c^17+831014*c^16+385741*c^15+175048*c^14+77684*c^13+33708*c^12+ 14290*c^11+5916*c^10+2398*c^9+950*c^8+365*c^7+132*c^6+42*c^5+14*c^4+5*c^3+2*c^2+c+1

Gn(8) = Fn(8)/(Gn(1)*Gn(2)*Gn(4)) = c^120+60*c^119+1770*c^118+34250*c^117+489375*c^116+5511102*c^115+50989975*c^114+ 398959395*c^113+2696755065*c^112+16009488760*c^111+84574656816*c^110+401877006480*c^109+1733107463040*c^108+6834914821260*c^107 +24811070182620*c^106+83371586399039*c^105+260615957165336*c^104+761188082391138*c^103+2085341983243658*c^102+ 5377303587067725*c^101+13092095906249710*c^100+30181158127871065*c^99+66047545613729349*c^98+137526242897521387*c^97+ 273055685652049512*c^96+517973316808768672*c^95+940463418168922896*c^94+1637134234590816416*c^93+2736605515930668712*c^92+ 4399030584560979688*c^91+6809393331711977377*c^90+10162926651591927936*c^89+14642200658326619088*c^88+ 20387237220949711442*c^87+27462099883710267563*c^86+35823271041713949254*c^85+45296178536637687081*c^84+ 55565997236718038571*c^83+66187135731800060711*c^82+76612820738925961694*c^81+86242488958463274826*c^80+ 94481112789965725168*c^79+100801976018285941440*c^78+104803422336957136740*c^77+106250977507865123996*c^76+ 105098793143557019895*c^75+101487983924396807096*c^74+95723317903066081340*c^73+88233057204728849436*c^72+ 79518917115832434169*c^71+70103821471231936142*c^70+60484437480620582867*c^69+51093714978195649483*c^68+ 42276335962975311453*c^67+34277623335577859252*c^66+27244489096635136284*c^65+21235681116184560728*c^64+ 16237990133298034432*c^63+12185127163725489840*c^62+8976499782758573072*c^61+6493890970072701905*c^60+4614878920815010176* c^59+3222581751710173440*c^58+2211880343460599616*c^57+1492636547692947392*c^56+990597359091017086*c^55+646699117035044471* c^54+415409562944557103*c^53+262617111655159022*c^52+163433639370729534*c^51+100144730020422499*c^50+60433299357506429*c^49+ 35923338854117473*c^48+21038627397895636*c^47+12141832138153068*c^46+6906542871283655*c^45+3872827682093776*c^44+ 2141241178405040*c^43+1167479848735120*c^42+627846072983732*c^41+333079003836572*c^40+174340431656069*c^39+90046721164189* c^38+45900127501242*c^37+23093412618276*c^36+11469308277609*c^35+5623447798913*c^34+2722207351629*c^33+1301145233716*c^32+ 614110737048*c^31+286228647553*c^30+131751286720*c^29+59896973608*c^28+26896860416*c^27+11931393376*c^26+5229116872*c^25+ 2264514156*c^24+969160031*c^23+409965164*c^22+171422294*c^21+70854302*c^20+28948921*c^19+11691330*c^18+4667641*c^17+1842415*c^16 +718975*c^15+277264*c^14+105560*c^13+39636*c^12+14680*c^11+5368*c^10+1944*c^9+698*c^8+248*c^7+84*c^6+28*c^5+8*c^4+1

/* Maxima function based on algorithm by Robert Munafo */
(%i2) GiveG(p):=
block(
/* if cc could not be computed then gives empty list */
[f:ifactors(p)],
/* ------------ prime numbers ------------------------------------- */
if (p=1) then g:G(p,0,c)
elseif primep(p) then g:G(p,0,c)/G(1,0,c) /* prime > 1 */
/* ------------ composite numbers ------------------------------------- */
/* PowerOfPrime, for example 9=3*3 */
elseif length(f)=1 then g:G(p,0,c)/G(p/2,0,c)
/* Product Of 2 different Primes */
elseif length(f)=2 and f[1][2]+f[2][2]=2 then g:G(p,0,c)*G(1,0,c)/(G(f[1][1],0,c)*G(f[2][1],0,c))
/* p=q*r*r, a prime times the square of another prime */
elseif length(f)=2 and f[1][2]+f[2][2]=3
/* (Zn(p)*Zn(r))/(Zn(q*r)*Zn(r*r)) */
then g:G(p,0,c)*G(r(p),0,c)/(G(p/r(p),0,c)*G(r(p)*r(p),0,c)),
return(g)
)$

/* New function giving G
t is temporary variable = product of G for (divisors of p) other then p */
GiveG[p]:=
block(
[f:divisors(p),t:1],
f:delete(p,f), /* delete p from list of divisors */
if p=1
then return(Fn(p,0,c)),
for i in f do t:t*GiveG[i],
g: Fn(p,0,c)/t,
return(ratsimp(g))
)$

/* degree of Fn is 2^(n-1)*/
d(n):=hipow(expand(Fn(n,0,c)),c);

makelist(d(n),n,1,10);
/* n = [1,2,3,4,5 ,6 ,7 ,8 ,9 ,10]
/* degree of Fn = [1,2,4,8,16,32,64,128,256,512] */
/* degree of G = [1,1,3,6,15,27,63,120] */
/* difference = [0,1,1,2,1 ,5 ,1 ,8] */
/*numberOfCenters=[1,1,3,6,15,27,63,120,252,495] */

makelist(2^(n-1),n,1,30);
[1,2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,131072,262144,524288, 1048576,2097152,4194304,8388608,16777216,33554432,67108864,134217728,268435456,536870912]

synonims:
Nucleus of Mu-atoms

Programs :

Multiprecision Polynomial solver
Deep Zoom by Jay R. Hill
WinSolve finds all zeros of a polynomial of any degree with either real or complex coefficients
Mandelbrot Root Explorer byJunichi Sato
Rootfinders by Hans D. Mittelmann

See also lists and images of centers :

centers of components computed with eigensolve by Steven Fortune (periods 1-13)
centers of components computed with MPSolve up to period 14
centers of components computed with MPSolve
centers and its symbolic sequences for periods 1-10 computed with Maxima bfallroots
centers of components computed with Maxima command bfallroots from new cpoly.lisp by Raymond Toy and qdynamics
centers of components computed with Maxima command allroots
centers of components computed with Maxima package polyroots
- old version
- new version (better)
centers of components computed with WEb solver
centers of components computed with GNU Octave
real centers of components computed by Jay R. Hill
MSS symbolic sequences of centers

References :

[Alvarez98a] G. Álvarez, M. Romera, G. Pastor, F. Montoya, "Determination of Mandelbrot set hyperbolic component centres", Chaos Solitons and Fractals, 9 (1998), 1997-2005. (PDF) - list centers up to 7 period with its sumbolic sequences and Myrberg's method for computing them
[Do1] V Dolotin , A Morozow : On the shapes of elementary domains or why Mandelbrot set is made from almost ideal circles ? list of centers up to 6 period
[Yo1] Young Hee Geum, Young Ik Kim : An improved computation of component centers in the degree-n bifurcation set. The Korean Journal of Computational & Applied Mathematics Volume 10 , Issue 1-2 (September 2002) Pages: 63 - 73 2002
[Yo2] Young Hee Geum, Young Ik Kim : A STUDY ON COMPUTATION OF COMPONENT CENTERS IN THE DEGREE- n BIFURCATION SET International Journal of Computer Mathematics, Volume 80, Issue 2 February 2003 , pages 223 - 232
centers of hyperbolic componenets
real component centers of Mandelbrot Set, period less than 16. by Jay R. Hill
Sci.fractal posts about DeepZoom with list of centers by Jay R. Hill
Sci.fractal posts about complex arithmetic with list of centers by Jay R. Hill
Professor Douglas C. Ravenel course
Wikibook

Main page

Autor: adammaj1-at-o2-dot--pl Adam Majewski Feel free to e-mail me. (:-)) About