Dynamic limits of sequences

In logic, negation can sometimes be interpreted as a function mapping 0 to 1, and 1 to 0.

This leads to issues when defining semantics, especially with recursion, as we like using Tarski-Knaster, but this supposes having interpretations that are all monotone.

In modal mu-calculus, for instance, one therefore restricts in general their attention to the positive fragment, in which each variable bound by a recursion operator shall appear under an even number of negation operators.

This relates to classic undergrad concepts in analysis: take a sequence with u_0 = 0, and u_{n+1} = 1 - u_n, then it will forever alternate between 0 and 1. But then, the two subsequences u_{2n} and u_{2n+1} converge -- but to different values, namely 0 and 1. These are the two accumulation points of the original sequence u_n, which does not converge.

And still, I'd say that it converges -- but to a dynamical system akin to a finite state machine.

One could define the limit of u_n as a finite state machine with two states 0 and 1 (one for each accumulation point), and transitions between each other, modeling the "+1" on indices.

Of course, this requires an adequate definition of such machines, and a notion of minimisation over them, as infinitely many representations could be given in general.

This also leads to some homology-like notion over sequences, which can then be studied from the point of view of non-convergence not only via the amount of accumulation points they have, but also via the dynamic relations between these accumulation points.

I remember an old exercise: consider n |--> exp(2*i*n*Pi) and show that the set of its accumulation points is dense in the unit circle. This already provides an example of sequence with a very "bad" homology.

What could we get from such a dynamic, generalised notion of limit for sequences?

I have focused on this example swapping in between 0 and 1, but obviously there should be some epsilon at play here.

Like: for every epsilon > 0, there is a rank N after which the sequence follows the dynamics of its generalised limit by being always at a distance smaller than epsilon from the relevant accumulation points.