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math185-s23:s:hexokinase:start [2023/04/14 07:46]
hexokinase [Fixed points of automorphisms of $\mathbb{D}$] 		math185-s23:s:hexokinase:start [2026/02/21 14:41] (current)
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 **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. **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 $\partial\mathbb{D}$. Maybe they are precisely those $f$ which have two fixed points in $\partial\mathbb{D}$.
  
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 **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 **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
-$$ u \pm \sqrt{ 1 - |u|^{-2} } $$+$$ u \left(1 \pm \sqrt{ 1 - |u|^{-2} } \right) $$
 where $u = \frac{ 1+\lambda }{ 2\overline{a} }$.\\ where $u = \frac{ 1+\lambda }{ 2\overline{a} }$.\\
 This completely resolves the unresolved question; This completely resolves the unresolved question;
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-{{ :math185-s23:s:hexokinase:9.pdf |}}+{{ :math185-s23:s:hexokinase:9.pdf |}}\\ 
 +{{ :math185-s23:s:hexokinase:10.pdf |}}
math185-s23/s/hexokinase/start.1681458389.txt.gz · Last modified: 2026/02/21 14:43 (external edit)

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