OK, my knowledge on this topic is completely superficial, so my notation will be a fudge. Nevertheless I am really interested in what some of the more clued up members have to say about this.

One can consider a certain type of "self-referential" expression which includes a complete copy of itself as a part of its structure. A way of expressing this in some kind of notation is to denote a sentence S, where

S := S + C

And C being constant. From my understanding, variations of these ideas are studied in dynamical systems, where a system is iterated and its behaviour at infinity studied.

Anyway, what happens if you construct an expression of this form so that its powers of "self-reference" diminish after multiple loops? For example:

S := (1/2)S + C

Do you consider this expression to have basically no self-referential nature after a large amount of loops? Does it even make sense for different levels of complete self-reference to exist?

I'm afraid this is a huge jumble of ideas, and the very last thing I scribbled on my notebook before going to sleep after about 40 solid hours of on-and-off studying

In dynamics, we sometimes use statements like this, although often more complex to compute invariant sets of iterated function systems. The solution of your particular problem is inevitably dependant on the space you are iterating in: in the riemann sphere we have s={infinity} (invariant under all polynomials) and zero if c=o.

However if your space is toral (or for that matter any compact manifold) the right choice of c (potentially a vector) will yield a more interesting set, perhaps the whole manifold.

(Don't quote me on that, although I know that for the surface of revolution of the tractrix [half psuedosphere] this does give an s1 subset for the right coordinate choice- potentially more may be acheived for other hyperbolic 2 manifolds [by pick's theorem {awesome deep result-check it out!}, if it constitutes a conformal isomorphism it will preserve poincare distance])

This brings me neatly onto the next usage: topology and quotient spaces. So long as your transformation is invertible, then it constitutes a group action on your space, this will most likely be trivial (collapsing it to a point), however in some cases (eg s':=s+c on R) we may identify by equivalence relation all orbits (sets mapped to and from) of points in our space creating a new one represented as a subset of our old space (by monodromy). The example I gave gives s1. More interesting examples require more group elements, such as the fuschian tori with genus>1 on the hyperbolic plane i showed you, those need a subgroup of mobius maps to work.

In logic, that first statement and its set theoretic analogues are used to define infinities. Your second statement has no appplication that I know of.

There, that's wasted some revision time nicely...

Damn, I wish I knew all this stuff :) I love how maths seems to get all introspectively and emo-y at a higher level, instead of just proportionally longer expressions and stuff.