# Iterated Function Sequences Accumulation: Help!

1. Apr 4, 2009

### aridneptune

1. The problem statement, all variables and given/known data
We are given three contractions which generate the Sierpinski right triangle:

A0 = $$\frac{1}{2}$$ <x , y>

A1 = $$\frac{1}{2}$$ <x-1 , y> + <1 , 0>

A2 = $$\frac{1}{2}$$ <x , y-1> + <0 , 1>

We are asked to find the points on which the sequence

(A2$$\circ$$A1)n(<x0 , y0>) ) -- that's (A1 COMPOSITE A2)n

accumulates.

3. 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: Apr 5, 2009
2. Apr 4, 2009

### xaos

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. Apr 5, 2009

### aridneptune

Sorry -- correction it's (A1 COMPOSITE A2)n (fixed above). Increasing n increases both applications of A1 and of A2.

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