Iterated Function Sequences Accumulation: Help

[aridneptune]

Homework Statement

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.

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!

 

[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.

 

[aridneptune]

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