# Convergence of Affine Transformations

Hi,

I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a_n} is a set of affine transforms and the sequence:

a_n (a_{n_1}(...a_1)...) converges to something as n->infinity. (Preferably another affine transform).

I know my question is bit vague, most of what I've been able to dredge up so far seemed to be based on probability, but I'm not necessarily looking at probabilistic results.

Thanks.

EDIT (Second edit): I had put in a restriction here - but on second thought I was wrong about the restriction so the question still stands as is. Sorry.

• Could you give us some more clues as to what you're looking for. Any convergent sum is an example of this phenomenon (since x --> x+a_i is an affine map, the sum \sigma a_i converges if and only if the corresponding sequence of translations does.) Similarly, any infinite product is an example. – David E Speyer Oct 16 '09 at 5:34
• Well I am not entirely sure what I am looking for yet, but I do see your point. I am pretty sure I can derive some constraints for which the things converge and a_ix + b_i don't all have a_i = 1 or b_i = 0 - but I am sure other questions could be asked i.e. Is there a subgroup which any members will converge in this fashion when iterated? Are there non-obvious conditions for convergence? Is there anything about what happens to the image of such transforms in general? Can I have a puppy? :) I am working on narrowing down what I need, but at the moment I am collecting puzzle pieces. – streklin Oct 16 '09 at 12:41