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--- math185-s23:s:hexokinase:start [2023/04/14 06:51]
hexokinase [Fixed points of automorphisms of $\mathbb{D}$]
+++ math185-s23:s:hexokinase:start [2026/02/21 14:41] (current)
@@ Line 44: / Line 44: @@
 **Fixed point in $\overline{\mathbb{D}}$:** the Brouwer fixed point theorem implies that every $f \in \operatorname{Aut}(\mathbb{D})$ has a fixed point in $\overline{\mathbb{D}}$ (once $f$ has been extended holomorphically to an open superset of $\overline{\mathbb{D}}$). It would be interesting to see a complex-analytic proof, especially since I don't know how to prove the Brouwer fixed point theorem.
-**Unresolved question:** which $f \in \operatorname{Aut}(\mathbb{D})$ have no fixed point in $\mathbb{D}$?\\
+**<del>Un</del>resolved question:** which $f \in \operatorname{Aut}(\mathbb{D})$ have no fixed point in $\mathbb{D}$?\\
 Maybe they are precisely those $f$ which have two fixed points in $\partial\mathbb{D}$.
@@ Line 51: / Line 51: @@
 Rouché's theorem might give a little more info.
+**Solution:** Just realized that, whenever $\lambda \in \partial\mathbb{D}$ and $0 \neq a \notin \partial\mathbb{D}$, the fixed points of $\lambda B_a$ are
+$$ u \left(1 \pm \sqrt{ 1 - |u|^{-2} } \right) $$
+where $u = \frac{ 1+\lambda }{ 2\overline{a} }$.\\
+This completely resolves the unresolved question;
+I might use this to write up a cleaner version of this post at some point.\\
 =====Homework solutions=====
 {{ :math185-s23:s:hexokinase:1.pdf |}}\\
@@ Line 58: / Line 63: @@
 {{ :math185-s23:s:hexokinase:7.pdf |}}\\
 {{ :math185-s23:s:hexokinase:8.pdf |}}\\
-{{ :math185-s23:s:hexokinase:9.pdf |}}
+{{ :math185-s23:s:hexokinase:9.pdf |}}\\
+{{ :math185-s23:s:hexokinase:10.pdf |}}

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