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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.


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.

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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 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
up vote 1 down vote accepted


could you perhaps specify what kind of space your transformations are acting on? Before you do that, let me try to still share some things...

If it's a Riemannian (sub)manifold, e.g., Euclidean plane, then your problem fits well within the framework of dynamical systems, in particular "discrete-time" dynamical systems as your transformations are indexed by a countable set. Depending on the transformations, you might end up having an attracting set, and the limiting operator would amount to a kind of "dynamical projection" of the entire space to that attracting set. If transformations are all equal, i.e., a_k = a_l, for all k,l, then you have an "affine, time-invariant (autonomous) system". If not, you have an "affine, time-varying (non-autonomous) system".

Most of dynamical systems will not be phrased exactly as you presented it, rather, the sequence of transformations will be generated as a solution of a difference/differential equation, especially at the introductory level. The language you are using is more common in ergodic theory, which I think is the appropriate setting for types of questions you are asking. It deals with limiting processes for (semi)groups of operators, however, the operators in question are often considered to be linear (they are composition operators on spaces of functions/distributions). Perhaps more advanced texts do generalize to non-linear operators. Additionally, reapplication of affine transformation (if it's the same one), translates to a reapplication of a linear operator + a series generated by reapplication of linear operator to the offset vector, in a handwavy way. :) I therefore believe there is hope for your problem within the context of ergodic theory. As an intro text, perhaps you'd want to look into Silva: Invitation to ergodic theory (recently published by AMS). For a more advanced treatment, you'd have to look for something more advanced, e.g., Petersen to start with, but perhaps going to Cornfeld, Fomin, Sinai (I have still to even parse through that).

Hope this helped.

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Thanks! That does help! As for the kind of space it is Euclidean however I am not exactly creating a dynamical system (and yet I am yay!) Basically I already have a dynamical system in the form a fuzzy cellular automata and am attempting to figure out what happens to its dynamics (especially the fixed points) when they are transformed by repeated (possibly infinite) affine transformations (which may or may not be distinct). I'll take a look at ergodic theory and where that leads based on your answer. Again - thanks for the help! – streklin Oct 23 '09 at 12:35

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