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MathGroup Archive 2004

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Re: tetrahedral Siegel Disk Julia map

  • To: mathgroup at smc.vnet.net
  • Subject: [mg52413] Re: tetrahedral Siegel Disk Julia map
  • From: Roger Bagula <tftn at earthlink.net>
  • Date: Fri, 26 Nov 2004 01:04:49 -0500 (EST)
  • References: <co4ebr$lbo$1@smc.vnet.net>
  • Reply-to: tftn at earthlink.net
  • Sender: owner-wri-mathgroup at wolfram.com

        http://www.math.harvard.edu/~ctm/gallery/menu.html

A gallery of tetrahedral / K3 Siegel disk dynamics by Dr McMullen


        Dynamics on K3 surfaces

K3 movie <http://www.math.harvard.edu/%7Ectm/gallery/movies/k3.movie.gif>
K3 surface <http://www.math.harvard.edu/%7Ectm/gallery/k3/k3.surf.gif>
Tame <http://www.math.harvard.edu/%7Ectm/gallery/k3/k3.tame.gif>
Wilder <http://www.math.harvard.edu/%7Ectm/gallery/k3/k3.wild.gif>
Ergodic <http://www.math.harvard.edu/%7Ectm/gallery/k3/k3.ergo.gif>
Stable manifold 
<http://www.math.harvard.edu/%7Ectm/gallery/k3/stableman.gif>
Tame blowup 
<http://www.math.harvard.edu/%7Ectm/gallery/k3/k3.tameblowup.gif>
Poncelet's theorem 
<http://www.math.harvard.edu/%7Ectm/gallery/k3/conics.gif>
Poncelet Java 
<http://www.math.harvard.edu/%7Ectm/gallery/poncelet/index.html> 
(Schwartz) <http://www.math.umd.edu/%7Eres>
Dynamics on a (2,2) curve 
<http://www.math.harvard.edu/%7Ectm/gallery/k3/curve22.gif>


Roger Bagula wrote:

>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
>
>  
>

-- 
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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