
<<special_points

(*Here are some plotting routines*)
SetOptions[Graphics,AspectRatio->Automatic]

(*plotting a complex number*)
pt[z_]:=Point[{Re[z],Im[z]}]

(*plotting a segment*)
sg[{z_,w_}]:=Line[{{Re[z],Im[z]},{Re[w],Im[w]}}]




(*Check the action of A[1] on the fixed points of B[-4].
A[1] should map the fixed points of B[-4] to the
fixed points of B[6], in a way which respects the
norm-ordering.  The smaller norm fixed point gets mapped
to the smaller norm fixed point*)

ActionOnFixedPoints1[]:=(
h1=FixedPoints[B[-4]];
h2=Mobs[A[1],h1];
h3=FixedPoints[B[6]];
{h2,h3})

(*Check the action of A[0] on the fixed points of A[3]].
A[0] should map the fixed points of A[3] to the
fixed points of A[-10], in a way which respects the
norm-ordering.  The smaller norm fixed point gets mapped
to the smaller norm fixed point*)

ActionOnFixedPoints2[]:=(
h1=FixedPoints[A[3]];
h2=Mobs[A[0],h1];
h3=FixedPoints[A[-10]];
{h2,h3})











SpecialPointPlot1[]:=(
aa=FixedPoints[A[0]];
p0=pt[aa[[1]]];
p1=pt[ControlAlphaMinus[0]];
p2=pt[AlphaMinus[0]];
p3=pt[PlanePA[0]];
p4=pt[AlphaPlus[0]];
p5=pt[ControlEvenAlphaPlus[0]];
p6=pt[aa[[2]]];
q0=sg[FixedPoints[A[0]]];

Show[Graphics[{
Point[{0,0}],
RGBColor[0,1,1],
q0,
RGBColor[1,1,0],
p0,
RGBColor[1,.5,0],
p1,
RGBColor[1,0,0],
p2,
RGBColor[1,0,1],
p3,
RGBColor[0,0,1],
p4,
RGBColor[0,.5,1],
p5,
RGBColor[0,1,1],
p6}]]
)


SpecialPointPlot2[]:=(
aa=FixedPoints[B[0]];
p0=pt[aa[[1]]];
p1=pt[ControlBetaMinus[0]];
p2=pt[BetaMinus[0]];
p3=pt[PlanePB[0]];
p4=pt[BetaPlus[0]];
p5=pt[ControlBetaPlus[0]];
p6=pt[aa[[2]]];
q0=sg[FixedPoints[B[0]]];

Show[Graphics[{
Point[{0,0}],
RGBColor[0,1,1],
q0,
RGBColor[1,1,0],
p0,
RGBColor[1,.5,0],
p1,
RGBColor[1,0,0],
p2,
RGBColor[1,0,1],
p3,
RGBColor[0,0,1],
p4,
RGBColor[0,.5,1],
p5,
RGBColor[0,1,1],
p6}]]
)


SpecialPointPlot3[]:=(
aa=FixedPoints[A[1]];
p0=pt[aa[[1]]];
p1=pt[ControlAlphaMinus[1]];
p2=pt[AlphaMinus[1]];
p3=pt[PlanePA[1]];
p4=pt[AlphaPlus[1]];
p5=pt[ControlAlphaPlus[1]];
p6=pt[aa[[2]]];
q0=sg[FixedPoints[A[1]]];


Show[Graphics[{
Point[{0,0}],
RGBColor[0,1,1],
q0,
RGBColor[1,1,0],
p0,
RGBColor[1,.5,0],
p1,
RGBColor[1,0,0],
p2,
RGBColor[1,0,1],
p3,
RGBColor[0,0,1],
p4,
RGBColor[0,.5,1],
p5,
RGBColor[0,1,1],
p6}]]
)



(*this plot shows  p1=A_0(alpha^{3}^0) and p2=alpha_{-10}^0.
p1 is the center of the blue x and p2 is the red dot.
The blue segment with the red dot is alpha(-10).  The 
segment A_0(alpha(3))) coincides with alpha(-10) *)

SpecialPlot4[]:=(
Show[Graphics[{
Point[{0,0}],
Circle[{0,0},1],
RGBColor[1,0,0],
sg[FixedPoints[A[-10]]],
sg[FixedPoints[HA[-10]]],
pt[PlanePA[-10]],
RGBColor[0,0,1],
sg[Mobs[A[0],FixedPoints[A[3]]]],
sg[Mobs[A[0],FixedPoints[HA[3]]]]}]]
)
