Iteration basics

[bo198214]

What if we contruct a 2 variable function F(t,x) where t is iteration of f(x) , x - argument of f(x) and then parametrize, for example:

It could be also possible to make t an imaginary part of a complex number, while x would be real part, so that:

\( z = x+it = h(y) + I* g(y) \)

So then F(t,x) becomes F(z) if functions h(y) and g(y) satisfy Euler conditions for derivatives of complex number functions.

What I wanted to ask, is , has this been considered, where could I read about it and does it make any sense at all to look at such functions?

Never heard about it, and have no idea whether this is bringing interesting considerations.