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math185-s23:s:hexokinase:start [2023/04/14 06:38]
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. |
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| **Unresolved question:** which $f \in \operatorname{Aut}(\mathbb{D})$ have no fixed point?\\ | **<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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| **Thoughts:** The function $f = \lambda B_a$ is nonconstant whenever $a \notin \partial\mathbb{D}, \lambda\neq0$. It's meromorphic on $\hat{\mathbb{C}}$. Let $a \neq 0$; then the quadratic formula shows that $f$ has exactly two fixed points with multiplicity. They may be equal: let $\lambda \in \partial\mathbb{D}$; then $f$ has a double fixed point iff $ \frac{ |1+\lambda| }{ 2|a| } = 1$, and the double fixed point is $\frac{ 1+\lambda }{ 2\overline{a} }$. (Note that $\arg \lambda = 2 \arg (1+\lambda)$, which is a fact about inscribed and central angles in circles.) Let $\lambda \neq -1,1$; those $a$ solving the equation form a circle centered at $0$ with radius $\tfrac{1}{2} | 1+\lambda | < 1$.\\ | **Thoughts:** The function $f = \lambda B_a$ is nonconstant whenever $a \notin \partial\mathbb{D}, \lambda\neq0$. It's meromorphic on $\hat{\mathbb{C}}$. Let $a \neq 0$; then the quadratic formula shows that $f$ has exactly two fixed points with multiplicity. They may be equal: let $\lambda \in \partial\mathbb{D}$; then $f$ has a double fixed point iff $ \frac{ |1+\lambda| }{ 2|a| } = 1$, and the double fixed point is $\frac{ 1+\lambda }{ 2\overline{a} }$. (Note that $\arg \lambda = 2 \arg (1+\lambda)$; this has a straightedge-and-compass proof.) Let $\lambda \neq -1,1$; those $a$ solving the equation form a circle centered at $0$ with radius $\tfrac{1}{2} | 1+\lambda | < 1$.\\ |
| From this we can see that for any $a \in \mathbb{D}$ there exists a unique rotation $(z \mapsto \lambda z)$ such that $\lambda B_a$ has a double fixed point on the unit circle (and no other fixed points). | From this we can see that for any $a \in \mathbb{D}$ there exists a unique rotation $(z \mapsto \lambda z)$ such that $\lambda B_a$ has a double fixed point on the unit circle (and no other fixed points).\\ |
| | Rouché's theorem might give a little more info. |
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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 |
| | $$ u \left(1 \pm \sqrt{ 1 - |u|^{-2} } \right) $$ |
| | 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===== | =====Homework solutions===== |
| {{ :math185-s23:s:hexokinase:1.pdf |}}\\ | {{ :math185-s23:s:hexokinase:1.pdf |}}\\ |
| {{ :math185-s23:s:hexokinase:7.pdf |}}\\ | {{ :math185-s23:s:hexokinase:7.pdf |}}\\ |
| {{ :math185-s23:s:hexokinase:8.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 |}} |
