air on swinging back and forth

The core of the iteration here is a process, seeded at s=1, such that for each cycle of the iteration we have:

S = f(z/s)
s = f(S/z)

It doesn’t actually have to be division, just some inverse process, and because I am fond of giving bright and strange names to things, I propose that we call these functions hā.ʻukē.ʻukē

hā.ʻukē.ʻukē: vi. To swing back and forth with a bang or clatter; to click as the teeth; to bounce back and forth, as breasts.