What makes an affine map repeat?

Some affine maps have certain points that when plugged in, give themselves. Causing an infinite loop. Take FCF for example:

F1F( -1/2x+1/2, 1/2y).......F1F( 1/3, 0) ---> ( 1/3, 0)
F2C( -1/2x+1, -1/2+1/2).....F2C(2/3, 1/3)--->(2/3, 1/3)
F3F( -1/2x+1/2, 1/2y+1/2)...F3F(1/3, 1) --->(1/3, 1)

It is interesting to note that the repeating points in F1 and F3 are the end points of the straight line between the two sections. It is also interesting to note that if you took the point (1/3, 1/4) and put it through F1F it would approach the point (1/3 , 0) since 1/3 would stay 1/3 and 1/4 would become 1/8 which would become 1/16 getting infinitely smaller. These point seem something like the attractors I read about earlier. They help when trying to prove if two sections are connected.

Sierpinski Relatives: Straight Lines

Using the relative FCF as an example I was shown a simple way to prove there was a vertical line through S1 (bottom right) and S3 (top right). In order for this to work the x-components had to match, and the y values had to go from [0,1/2] for F1 and [1/2,1] for F3.

F1F(-1/2x+1/2, 1/2y) F3(-1/2x+1/2, 1/2y +1/2)

As you can see these two maps satisfy both properties. Since F2 was not used in this proof at all, an relative with F as the map for F1 and F3 will have a straight line through S1 and S3.

We can use almost the same technique to prove there is a horizontal line through S1 and S2 (bottom left). Instead the Y components must match and the X values must got from [0,1/2] for F1 and [1/2,1] for F2. The relative with the form EE_ will have a horizontal line through S1 and S2.

F1E(1/2x, -1/2y+1/2) F2E(1/2x+1/2, -1/2y+1/2)

-1/2y+1/2 = -1/2y+1/2 The y values match

1/2x=x ... x=0
1/2x+1/2=x ... x=1, x values go from [0,1]