I'm trying to work on a theory of "compressive functions"/"compressor recursions"—functions which, when applied recursively to a domain (which I'll probably restrict to being discrete) bring all elements to a certain finite subset of the original domain. It would be sort of a discrete maths variant of the idea of attractors in differential equations.
An example would be adding up all the digits, with a finite "compressed" subset of single digit numbers. Also if the Collatz conjecture is true, then the "compressed" subset would be the {1, 2, 4} cycle. In both examples given, the whole infinite domain doesn't actually "compress" at all with finite iterations. Would I be able to adapt the epsilon-delta formalism/definition for limits like this? which avoids the infinite domain: For any given finite subset (epsilon), there exists a number (delta) such that the function, applied delta times to the subset epsilon, compresses it into the "compressed" subset.
For summing the digits, the "compressed" subset is all single digit numbers, and epsilon is any arbitrary finite subset of N, and there is always a delta since summing the digits makes a number smaller. I probably explained it poorly, I'm willing to re-explain any part. Just trying to see if my definition makes sense.
The significance of this is that if through working on this, I can come up with an equivalent definition/equivalent set of conditions (or one that implies the original), then it could make proving the Collatz conjecture easier. I know, it's unrealistic, but I'm doing this mostly for fun, and just want to make sure that I'm still doing actual rigorous work.
You will probably be happier adapting the $\varepsilon-\delta$ definition of a limit at infinity...
Let $f^{(m)}$ is the $m$-times iterated application of $f$ and $C$ be the "compressed subset". For any finite $U \subset \Bbb{Z}$, there exists an $N = N(U)$ such that for all $n > N$ and all $u \in U$, $f^{(n)}(u) \in C$.