04/19/2023, 12:45 AM
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
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