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MathGroup Archive 2004

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tetrahedral Siegel Disk Julia map

  • To: mathgroup at smc.vnet.net
  • Subject: [mg52380] tetrahedral Siegel Disk Julia map
  • From: Roger Bagula <tftn at earthlink.net>
  • Date: Thu, 25 Nov 2004 05:49:55 -0500 (EST)
  • Reply-to: tftn at earthlink.net
  • Sender: owner-wri-mathgroup at wolfram.com

Siegel disks don't just happen in complex dynamics of quadratics.
You can set this type of "motion" going on other Riemannian surfaces
as Dr. McMullen suggested in his paper on K3 surfaces
using an tetraheral implicit surface and a Salem based irrational number.
In this simulation an Siegel disk is located on a Riemannian
tetraheral surface.

Clear[x,y,a,b,s,f,g,a0,t]
(*tetrahedral Siegel Disk Julia map*)
(* idea based on McMullen K3 ( tetrahedral) surface Siegel disk dynamics*)
z=x[n-1,t]+I*y[n-1,t]
z4=ComplexExpand[z^4-2*Sqrt[3]*I*z^2+1]
(* Riemannian Tetrahedron polynomial from Elliptic Curves, McKean and Moll,
  p22, Ellipical invariants of Platonic solids*)
(* j[z]=(z^4-2*Sqrt[3]*I*z^2+1)^3/(z^4+2*Sqrt[3]*I*z^2+1)  *)
f[n_,t_]=Re[z4]
g[n_,t_]=Im[z4]
gm=N[(1+Sqrt[5])/2];
a=Cos[2*Pi*gm];
b=Sin[2*Pi*gm];
digits=1500;
x[n_,t_]:=x[n,t]=x[n-1,t]*a-y[n-1,t]*b+f[n,t]/4
y[n_,t_]:=y[n,t]=x[n-1,t]*b+y[n-1,t]*a+g[n,t]/4
x[0,t_]:=0.27/(1+t/2);y[0,t_]=0.01/(1+t/2);
a=Flatten[Table[Table[{x[n,t],y[n,t]},{n,0, digits}],{t,1,10}],1];
ListPlot[a, PlotRange->All]
Respectfully, Roger L. Bagula

tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
alternative email: rlbtftn at netscape.net
URL :  http://home.earthlink.net/~tftn


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