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Let $\mathbb{D} = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 < 1 \}$ be the unit disk.

We say a function $f : \mathbb{D} \rightarrow \mathbb{D}$ is a winner if it satisfies the following:
1) it is a homeomorphism
2) if $f(x) = y$, then $f(y) = x$
3) the only fixed point is the origin $(0,0)$

It is easy to give at least one example of a winner function: the "point reflection" centered at the origin (i.e., a 180-degree rotation). Are there other examples?

As pointed out by Bill Johnson (see below), winners are invariant under conjugation by origin-preserving homeomorphisms. This raises the following question: is there another "conjugacy class" of winners?

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Do you mean "reflection"? I would have thought you meant rotation through 180 degrees – Yemon Choi Nov 1 '11 at 20:23
Yes, a 180-degree rotation is the same thing... – expmat Nov 1 '11 at 20:26
Oh, I see, I was brought up to use the word reflection for a reflection in a hyperplane (or mirror) – Yemon Choi Nov 1 '11 at 20:37

Condition (2) says $f^2=I$. Conjugate your reflection with a homeomorphism that fixes zero and does not commute with your reflection.

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That's correct. But are these the only ones? – expmat Nov 1 '11 at 20:44
@expmat: your original question just asked for an example which was not rotation through 180 degrees – Yemon Choi Nov 1 '11 at 20:48
@Bill: Excuse me for my ignorance but can you give me examples of such homeomorphisms fixing the origin but not commuting with my reflection? – expmat Nov 1 '11 at 20:52
Map one to minus one and minus one to any point on the unit circle other than one. Extend to any homeomorphism that fixes the origin. – Bill Johnson Nov 1 '11 at 21:09
Thanks! Is this the only "conjugacy class"? – expmat Nov 2 '11 at 19:37

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