Here is an applet to explore the function $$\phi(z) = \frac{z - \lambda}{1 - z \bar{\lambda}}$$ for $| \lambda | \leq 1$. In the bottom left is the unit disc with Darth Vader super imposed on it. On the top is also the unit disc. For each point $|z| < 1$, the applet determines the color of $\phi(z)$ and and then colors $z$ that color.

The unit disc on the bottom right allows you to adjust $\lambda$. Just click anywhere in the unit disc and $\lambda$ will change appropriately.

Note that this demonstrates that $\phi$ maps the unit disc to itself.

Click here for a proof.

The proof I'm a little confused on what your first claim is. I see you've built the inverse and demonstrated it's a right inverse (technically we'd need to show it's also a left inverse, but it's pretty trivial so we can grant it's a true inverse) but then your claim is that it shows the original function only maps into the disc is that right? I believe your argument is sound but that leap in logic is losing me. Maybe I'm just tired.

ReplyDeleteThe rest seems sound though.

PS: The app doesn't seem to load for me for some reason. Just shows a P in the upper left corner, which maybe means I should click something but not familiar with it.

The key idea is that $\phi_{-\lambda}:D \rightarrow C$, so $\phi_{-\lambda}^{-1}(D) \subseteq D$. (I'm not sure if LaTeX will render in the comments.)

DeleteThe P in the corner just indicates that it's loading, but that shouldn't take more than a few seconds. Is your Java up to date?

Curious. Apparently Firefox doesn't like it either, getting the same loading screen perpetually.

DeleteOh well, thanks for trying.

DeleteI updated the pdf to include your proof. Thanks!