Iterated Function Sequences Accumulation: Help!

  • #1

Homework Statement


We are given three contractions which generate the Sierpinski right triangle:

A0 = [tex]\frac{1}{2}[/tex] <x , y>

A1 = [tex]\frac{1}{2}[/tex] <x-1 , y> + <1 , 0>

A2 = [tex]\frac{1}{2}[/tex] <x , y-1> + <0 , 1>

We are asked to find the points on which the sequence

(A2[tex]\circ[/tex]A1)n(<x0 , y0>) ) -- that's (A1 COMPOSITE A2)n

accumulates.

The Attempt at a Solution



Not quite sure how to approach this problem at all. I've figured that A1n takes any <x , y> to <1 , 0>, and that A2n takes any <x, y> to <0, 1>. So my first instinct was to say that iterating A2 n times on an n-iteration of A1 would just converge to <0 , 1>. However, I'm fairly sure this is incorrect. But how can this system accumulate on >1 point?

Any ideas/help would be greatly appreciated!
 
Last edited:

Answers and Replies

  • #2
179
4
my first thought is to write the n-th iterators in terms of <x0,y0>, where the n-th iteration of A1 with be the initial point for the n-th iteration of A2.
 
  • #3
Sorry -- correction it's (A1 COMPOSITE A2)n (fixed above). Increasing n increases both applications of A1 and of A2.
 

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