Summary

Long description

Parameter of quadratic map

${\displaystyle c=-0.031110+0.79111*i}$

It is a center of period 9 hyperbolic component.

It means that critical point z=0 has period 9 under complex quadratic polynomials.

Period 9 cycle

Here is superattractive orbit of critical point :

z0 = 0.000000000000000 +0.000000000000000 i

z1 = -0.031467614571113 +0.790703205755478 i

z2 = -0.655688963396307 +0.740940118317765 i

z3 = -0.150531856784133 -0.180949310481546 i

z4 = -0.041550427627981 +0.845180477136670 i

z5 = -0.744071215468013 +0.720467985259779 i

z6 = 0.003100241332648 -0.281455773240590 i

z7 = -0.110675355365251 +0.788958044112452 i

z8 = -0.641673375655634 +0.616066781954641 i

z9 = 0.000738826626237 +0.000075902543203 i

abs(z0-z9) = 7.427152749886263*10^-4

Points are marked on image with white points

Period 3 cycle

In the Julia set there is a 3-period repelling orbit :

${\displaystyle {\begin{aligned}z_{3,0}=-0.055417986719313-0.165801721382951i\\z_{3,1}=-0.055886672132640+0.809080000942757i\\z_{3,2}=-0.682954742374583+0.700269628271949i\\\end{aligned}}}$

Here absolute value of multiplier is > 1 ( numerical computations give 1.11).

This points are the landing points for rays listed below.

Parabolic orbit portrait

Orbit portrait consist of 3 list, each has 3 angles :

${\displaystyle {\mathcal {P}}=\left\{\left({\frac {74}{511}},{\frac {81}{511}},{\frac {137}{511}}\right),\left({\frac {148}{511}},{\frac {162}{511}},{\frac {274}{511}}\right),\left({\frac {296}{511}},{\frac {324}{511}},{\frac {37}{511}}\right)\right\rbrace }$

These are angles of external rays landing of 3 points of parabolic cycle

Note that denominator of angles is :

${\displaystyle d=2^{9}-1=511}$

Period

Here are 5 values :

• orbit period = period of orbit from orbit portrait under complex quadratic map = p = 3
• period of external angle under doubling map = r = 3
• ray period = period of point z on ray under complex quadratic map = r*p = 9
• period of landing point under complex quadratic map = 3
• period of orbit of critical point = 9

How to use Mandel

Keys :

• e for externalray
• f for forward orbit
• s for description of angle and move to landing point

To check period of ray :

• go to the point on ray ( exterior of filled Julia set) near Julia set
• apply forward iteration using f key

To check period of landing point of ray :

• draw ray ( press e key and enter angle )
• press s key, enter the same angle and click OK
• * apply forward iteration using f key

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