Centers of components of Mandelbrot set





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

We will use second definition to compute centers.
because :
so multiplier of periodic orbit is equal to product of first derivative of Mandelbrot function at periodic points
ln(c) = 2n*Z0*Z1*.. *Zn-1

So ln(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 :



See also lists and images of centers :


References :


Main page


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