Question about the properties of iterated functions

[Shanghai46]

Hey!

So what you are referring to in the complex plane is known as "The immediate basin". So you have stuck to real analysis; and in such cases the immediate basin is an interval.

Note, that the only fixed points that have immediate basins are fixed points such that \(0 < |f'(\tau)| < 1\). As Tommy pointed out, if \(f'(\tau) = 0\), then this goes out the window. If \(|f'(\tau)| > 1\), then we'll know that if \(g(f(z)) = f(g(z)) = z\) that \(0 <|g'(\tau)| < 1\); which allows us to perform the same iteration tricks, but with the inverse case.

These cases are pretty damn well studied; but there's always new stuff popping up. The final case is the neutral case--which itself can be split into two cases. If \(f'(\tau)^n = 1\) for some \(n \in \mathbb{N}\)--than this is known as the parabolic case. If it doesn't satisfy this, it is an irrational rotation of the unit disk; and you'll have to study very advanced things like siegel disks.

All of these ideas have their counterparts in real analysis--and in my opinion it is even more complicated there! There's something inherently natural about using complex numbers for iteration theory (e.g: the Schroder function/Abel function/Julia function etc...). These things are, to say lightly, fairly awkward in real analysis in comparison to complex analysis.

Is there anything in specific you are interested in knowing? I'd be happy, as I'm sure Tommy would be, in answering questions. Excited to see what your method is! There are so many iteration techniques that it's getting ridiculous in recent years!  

Regards, James

Thanks for your reply. I'm currently working on my iteration technique, especially one thing : the interval restrictions of the starting number \(x_0\). I actually already presented it in my other posts if you're interested. I need to study my formula in order to get these restrictions done, and this post was one of my hypothesis about iterated functions that I took into account. I know that in math we need to work with proofs, even when we study it. But I mainly work with my intuition. Most of the time I'm right, but some times I'm wrong. When I'll be done I'll demonstrate everything, which is kinda already done since I have pieces of demonstration in my papers. 
If you want an example of hypothesis I make, is that for iterated functions that converge, and that \(f'(\tau)<0\)
The real iterations of it will give complex numbers. And my hypothesis, is that the argument of these complex numbers minus Tau change linearly as n increases. It equals 0 or -\(\pi\) when the iteration is an integer, and is plus or minus \(\pi/2\) when the iteration has a decimal part of one half. I haven't proved it, but with my equation, the examples I've tested, other iteration methods and my intuition, I think it's true.

But I'm kinda slow in terms of studying my equation cause I can't full time do it. Rn I'm 16, I'm in hollyday but I need to study for my exams. And it's going to be even worse next year (really freaking worse).