Hi,

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.

Iam 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