07/01/2022, 09:09 PM
I always like to relate things to the unit disk when doing these iterations.
The value \(\exp^{\circ it}(z)\) traces an almost elliptic path, similar to \(\lambda^it\) for \(\lambda\) the multiplier. Essentially the object looks very much like \(z \mapsto \lambda^i z\), but it's placed in this weird fractally shape (the immediate basin). If you map the immediate basin (using Schroder) to \(\mathbb{C}\), this comparison becomes directly evident. Fun fact, you can use this to prove that the immediate basin about the fixed point is simply connected. There exists a sequence of jordan curves which slowly approximate the boundary.
\[
\lim_{x\to\partial \mathcal{A}} \exp^{\circ it}(x)\\
\]
You can prove this is a jordan curve with some work, but it's doable. So picture a bunch of almost elipses which start to slowly get more and more fractally as \(x\) approaches the boundary, and more and more circular as x approaches the fixed point.
You can copy paste a lot from the mapping \(z \mapsto \lambda^i z\), thanks to the Schroder function. All in all, it's not something inherently that interesting though. The function \(\exp^{\circ it}(z)\) is definitely more interesting--especially for periodic, about the fixed point, solutions. Looking at the boundary of the immediate basin, you would never guess that it is a Jordan curve, and that the immediate basin is simply connected, but it is. Totally weird looking fractally shape that it is. This limit sort of lets you watch as a nice looking ellipse starts to look more and more fractally.
The value \(\exp^{\circ it}(z)\) traces an almost elliptic path, similar to \(\lambda^it\) for \(\lambda\) the multiplier. Essentially the object looks very much like \(z \mapsto \lambda^i z\), but it's placed in this weird fractally shape (the immediate basin). If you map the immediate basin (using Schroder) to \(\mathbb{C}\), this comparison becomes directly evident. Fun fact, you can use this to prove that the immediate basin about the fixed point is simply connected. There exists a sequence of jordan curves which slowly approximate the boundary.
\[
\lim_{x\to\partial \mathcal{A}} \exp^{\circ it}(x)\\
\]
You can prove this is a jordan curve with some work, but it's doable. So picture a bunch of almost elipses which start to slowly get more and more fractally as \(x\) approaches the boundary, and more and more circular as x approaches the fixed point.
You can copy paste a lot from the mapping \(z \mapsto \lambda^i z\), thanks to the Schroder function. All in all, it's not something inherently that interesting though. The function \(\exp^{\circ it}(z)\) is definitely more interesting--especially for periodic, about the fixed point, solutions. Looking at the boundary of the immediate basin, you would never guess that it is a Jordan curve, and that the immediate basin is simply connected, but it is. Totally weird looking fractally shape that it is. This limit sort of lets you watch as a nice looking ellipse starts to look more and more fractally.
