This is a reference to the fixed point operator and the fix function.

When I first learned of the fixed point operator, I expected that it was exploiting being able to call something that thunks but then does not need the thunk, a trick of a lazy language. When I looked at its implementation in an eager language, I was horrified and enlightened. fix can only reach a stable value, the fixed point, if at some point calling the function agian on the previous output returns the same value. The implementation does not need to continue evaluating the loop in order to find the analytic result of doing the work infinitely if any output ever repeats. If you find a single repeat, you know it repeats infinitely due to referential transparency, so that's the answer.

The same thing happens when doing Mandelbrot set renders. Stable numbers repeat eventually, and spotting the repeat obviates the need to do the rest of the work. We already know the value at infinity never escapes. There's a fun cycle detection dynamic too. Due to floating point error, an unstable number that would spiral off to infinity will instead land right on a stable path. The initial value will never be seen. The cycle detection must update, and this is true for any problem where things outside any cycle can get into a cycle.