Bounded-input bijections of infinite sequences
I am not a CS theory guy. But in ergodic theory this type of mapping is known as finitary isomorphisms. For example people considered if two Bernoulli sequences of the same entropy is finitarily isomorphic or not. For example (this is one-sided shift because it seems you are concerned with $P^{\mathbb{N}}$ rather than $P^{\mathbb{Z}}$):

A. Del Junco, “Finitary codes between one-sided Bernoulli shifts,” Ergodic
Theory Dynamical Systems, vol. 1, pp. 285–301, 1981.

P.S. I intend to leave this as a comment but I cannot due to lack of reputation. Let me know if it is completely off-topic then I will delete it.