Shigehiro Ushiki = Second Julia Page

• Movies for Symmetric Product maps.
• Movies for Symmetric Polynomial maps.
• Symmetric Polynomial maps for parameters in the left half.
• Symmetric Polynomial maps for parameters in the right half.
• Movies for Regular Polynomial maps
• Parametric variations of second Julia sets

Movies of second Julia sets of symmetric product maps in C2

Movies are stereo graphic

and in MPEG format(most movies are only for 4 seconds). Look at the right picture with your left eye and the left picture with your right eye. Fractal objects, "second Julia set" lives in complex two dimensional space, that is, in real four dimensional space. You are looking at the projection of the object to a real three dimensional space. The object is either rotating in the four dimensional space or changing their form according to the variation of the parameter. Rotation in the four-dimensional space is quite different from the rotation in the three dimensional space.

• For c=-2.0[movie1-1]248KB, Flying triangle.
• For c=-2.2[movie1-2]1.5MB, Cantor triangle.
• For c=0.0[movie1-3]577KB, Moebius band case.
• For c varied fron -2.0 to 0.05[movie1-4]8.3MB

Movies from symmetric polynomial maps in C2

Symmetric polynomial map was first studied by K.Uchimura. This map is defined by
X = x^2 + c y
Y = y^2 + c x
considred in the two dimensional complex space. Itsu second Julia set is computed by taking a generic point and plotting many points of its backward images. Each point of C^2 pas four points as its backward image( except critical points). In the following movies,4^10 points are computed and projected into the real three dimensional space and plotted in a stereographic way.

Double Mandelbrot set.

Parameters from left half of the double Mandelbrot set.

• For c=2.0[movie2-1]579KB, hypocycloid.
• For c=-2.01[movie2-2]1MB, Cantor set.
• For c=-1.99[movie2-3]650KB
• For c=-1.9[movie2-4]1.2MB, a space ship.
• For c=-0.6004851804+1.2496206i[movie2-5]1.2MB, a dendrite(?).
• For c=-0.5480566+0.95732930i[movie2-6]1.4MB, second Julia set for Douady's rabbit parameter.
• For c=-1.23606798[movie2-7]1.6MB, period two superattracting cycle.
• For c=-1[movie2-8]1.8MB, perod doubling parameter, parabolic basin.
• For c=1[movie2-9]317MB, saddle-node bifurcation, parabolic basin .
• For c=-2.2 to c=-0.2[movie2-10]5.7MB, metamorphose of second Julia sets.
  

  Parameters from right half of the double Mandelbrot set.
  

• For c=2[movie2-11]187KB, three cicles in the priodic complex lines and their backward images.
• For c=1.3[movie2-12]1.9MB, one dimensional Julia sets and their backward images.
• For c=2.5[movie2-13]897KB.
• For c=3.0[movie2-14]830KB, infinitely many copies of San Marco Juia sets.
• For c=4.0[movie2-15]613KB, Julia set in the invariant line and periodic lines are straight segemnts and their inverse images and the closure of them foem the second Julia set.
• For c=2.600485+1.24962166i[movie2-16]958KB,dendrite case.
• For c=2.54805+0.9573293i[movie2-17]991KB,Douady's rabbit case( one dimansional map has a superattractive cycle of period three ) .
• For c=3.236079[movie2-18]908KB, one dimensional dynamics has a superattractive cycle of period two.
• For c=2.8090-0.587785i[movie2-20]914KB,one dimensional map has a parabolic cycle of period 5.
• For c varied from 0.2 to 4.0[movie2-21]3.8MB, second Julia set transforms its shape from almost torus to dendrite via various fractal set.

Movies from regular polynomial maps in C2

Regular polynomial maps here are defined by
X = x^2 + a x + b y
Y = y^2 + c x + d y

• For a=0, b=1, c=-1, d=0.[Movie3-1][2.79MB]
• For a=-0.61607+0.455357i, b=1.205357-0.348214i, c=-0.455357-0.348242i, d=0.3482144-0.455357i.[Movie3-2][2.11MB]
• For a=0.7232144;0.133928i, b=0.8-0.9910714i, c=-1.098214+0.1i, d=0.0-0.455357i.[Movie3-3][2.43MB]
• For a=-0.61607;0.455357i, b=1.205357-1.3125i, c=-0.455357+1.473214i, d=0.348214387-1.258928i. [Movie3-4][2.86MB]
• For a=-0.6160714, b=0.401785, c=1.258928-0.40178579i, d=-1.5803570. [Movie3-5][1.83MB]
• For a=0.7232148+0.133928i, b=0.8, c=0.1i, d=0.2i. [Movie3-6][1.38MB]
• For a=0.12+0.74i, b=0.2i, c=0, d=0.25. This is a skew product case.[Movie3-7][1.81MB]

Movies from regular polynomial maps in C2
with varied parameters.

Regular polynomial maps (x,y) to (X,Y) here are defined by
u = x^2 + a x + b y
v = y^2 + c x + d y
X = p u + q v
Y = r u + s v

• Parameters are taken as a=-0.8, b=0.8, c=0.1i, d=0.2i,p=s=1,q=r=0. And c is varied to -2.9125, while other parameters are fixed. [Movie3-8][5.3MB]
• Initial parameters are same as the previous movie. They are moved to a=-0.8+0.723i, b=-1.8445, c=-0.134+0.1i,d=-0.416i.[Movie3-9][9.6MB]
• Initial parameters are sams as above. Parameters are varied to p=-1.259, q=0.884, r=-1.4196, s=1.0982.[Movie3-10][11MB]
• Initial parameters are a=-0.0268, b=-0.777, c=-0.7232-0.0248, d=0.13393i, p=s=1,q=r=0. They are varied by a : -0.0804i, b : -0.0804, c:-0.02678,d:0.026785i, p:-0.455357, q:-2.8125, r:3.0, s:-0.723214.[Movie3-11][4.7MB]
• Parameters are forgotten.[Movie3-12][7.7MB]