Properties of complex iterated functions. (with complex iterations)

Question

It's been more than a year I'm in touch with iterations of functions. A lot of my posts are about it. The problem being, it's not really a mathematical topic that gets a lot discussed, in books or in mathematical channels on YouTube (where all of my non - school/post high-school mathematical knowledge is from).

So I kinda have to do everything by myself, which is kinda hard, especially that (apparently) I don't have certain math knowledge to answer my questions.

So for context, I'll first explain what I'm talking about/findings, then the questions.

1. the formula

It's all about convergent iterated functions. The basic idea, was to have a formula, in order to calculate fractional iterations of a function, given a starting value. Here's the "recipe" :

• Take a real function $f$, which is differentiable on a certain domain that has an attractive fixed point $τ$ so that $0<|f'(τ)|<1$. (So $f$ differentiable at $τ$, and we will write $λ=f'(τ)$)
• Let $I$ be the biggest real interval such that :
  1. $I$ contains $τ$
  2. $f$ is monotonous on $I$
  3. $f(I)\subset I$
  4. $f$ is differentiable on $I$
• Then, $\forall x_0\in I, \forall α\in \mathbb{C},f^{\circ α}(x_0) = \lim_{n\to +\infty}{f^{\circ -n}((f^{\circ n}(x_0)-τ)λ^α+τ)}$

(Where $f^{\circ n}$ means the n-th iteration of $f$, $n$ being a positive integer. Also, $f^{\circ -n}$ means the n-th iteration of the inverse function of $f$ on $I$.)

Technically speaking, I should restrict this formula to real iterations, with strictly growing functions, otherwise it implies taking the inverse function of complex numbers. But in theory (correct me if I'm wrong), it still should work since usually, real functions can still have a defiened inverse function on complex numbers, and even if the result is indefiened for a certain iteration, it's isn't the matter.

2. Questions

1. Could this be extended to complex functions, with complex $τ$ and $λ$? I did some tests, for example, with the function $f(x)=i^x$, and it seems to work. But in that case, shall I add any more restrictions? For example, here "$I$" would be a zone in the complex plane. To avoid abuse in notations, I will call $J$ the zone of the complex plane that matches the original definition of $I$, but in the complex plane, even for real functions.(monotonous becomes injective here). Would it be conceivable to get a assymptote inside $J$ with its definition ? If yes, is it a problem ? If yes, what other restrictions shall I add? I don't know the potential cases that would break everything, so I'm unsure if my restrictions are enough. this question can also be asked with the original formula on the reals : is there a need for more restrictions to make it work all the time?

2. For all functions, real or complex, no matter the input $z_0$ (basically $x_0$, but taken in $J$) (also, the imput must obviously NOT EQUAL $τ$), or the iteration $α$, will all values of $f^{\circ α}(z_0)$ ALWAYS remain in $J$? I'm unsure, but I think it will. Since, in the formula (because of the limit and how it's constructed), we basically get to take as many iterations of the inverse functions of $f$ on $J$ of values that are infinitely close and all around $τ$ as we want. I think it would work. Cause, let's take a big $n$ for the limit, before repeating the inverse function, we have values extrely close to $τ$ all around it. We definitely are in not only $J$, but also in $f(J)$. Since $J$ is stable by $f$, I guess $f(J)$ would be stable by $f^{-1}$, and that either the values we will eventually always remain within $J$, or be undefined. Correct me if I'm wrong. If it's true, it would be extremely useful, since the iterated function would be the exact same no matter the original input, just shifted and stretched/squished. You take the 4D graph, and just move it so that $f^{\circ 0}=z_1$, your new input, strech in or out so that $f^{\circ 1}(z_1)=f(z_1)$, and you get the iterated function with the other input $z_1$. It would be technically the same function in a way.

3. If yes (2), are all values within $J$ reached by a certain iteration $α$? (A different input wouldn't matter here, if the previous point is true). Basically the surjectivity of the iterated function on $J$

4. If yes (2), would the iterated function be injective? (If (2) is correct, the starting value $z_0$ is irrelevant, as long as not equal to $τ$). So yeah, I know it won't be injective everywhere. Technically, it will have a complex period of $(2πi)/ln(λ)$. But on one of its period, would it be injective?

I hope all of these are true, that would make things easier, and being able to define the inverse of the iterated function.

I hope I was clear, explaining is not my cup of tea...These questions are just trying to seek new restrictions (if there isn't enough), and to confirm or disprove some useful properties of such iterated functions.

I tested the formula in many cases, and the properties seem to hold...but that doesn't ensure it will work all the time.

And before mentioning it, the Tetration Forum is dead...

Voilà !