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.



Convergence of Affine Transformations
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.


