# Proof of a fixed point theorem on the disk

There is a very nice fixed point theorem which I'd have liked to give to my students :

Let $n$, $m$ be two integers larger or equal to one. Let $B_n$ be the open unit ball in $\mathbb{R}^n$, and let $h$ be an homeomorphism of $B_n$ such that $h^m = id$. Then $h$ has a fixed point.

However, it seems that there is no proof of this general result which does not involve material far beyond their current reach. I was trying to design a simpler proof in dimension $2$, with nothing more evolved that Brouwer's fixed point theorem, invariance of domain, and perhaps (some avatar of) Jordan's curve theorem. I was thinking along these lines :

• Take a large enough closed ball $C$ in $B_2$. Then $K := \bigcup_{k=0}^{m-1} h^k (C)$ is connected.

• Show that complementary of $K$ has one connected component $\Omega$ which borders the disk. Take $K' := \Omega^c$.

• Show that $K'$ is a $h$-invariant simply connected compact. Use a version of Brouwer's fixed point theorem.

However, I have a hard time finding a suitable version of Brouwer's fixed point theorem. I could try to use Jordan-Schoenflies' theorem to conjugate the action of $h$ on $K'$ with the action of some homeomorphism of the closed unit disk, but then, I don't know much about the boundary of $K'$ (well, it is made of pieces of simple loops, but gluing them together seems messy).

Is there a nice and relatively elementary (but maybe completely different) way of proving the result I want ? If there isn't, how can I frame the argument above and where can I find suitable versions of Brouwer or Jordan's theorems to make things as painless as possible ?

This question has a very good answer on the MO counterpart Finite-order self-homeomorphisms of $$\mathbf{R}^n$$ by Nicholas Kuhn:
Let $$T$$ be a homeomorphism of period $$r$$ on Euclidean space En. The classical result of P. A. Smith is that $$T$$ has a fixed point if $$r$$ is prime [Ann. of Math. (2) 35 (1934), 572–578] or a prime power [Amer. J. Math. 63 (1941), 1–8; MR0003199]. The author settles a question of long standing with the following theorem: If $$r$$ is not a prime power, then there exists a triangulation $$μ$$ of $$E^{9r}$$ and a homeomorphism T of period r on $$E^{9r}$$ without a fixed point, such that $$T$$ is simplicial over $$μ$$. The construction is a modification of an example due to Conner and Floyd [Proc. Amer. Math. Soc. 10 (1959), 354–360; MR0105115].