TITLE: Homeomorphisms of the Open Disk
QUESTION [5 upvotes]: Does there exist a homeomorphism $\phi$ of the open unit disk in the plane such that $\phi$ has no fixed point but there exists $n$ such that the $n$-fold composition $\phi^n$ is the identity?
(To save some of you the bother of asking what I've done so far: nothing. Seems clear the answer is no; it's also clear to me that I simply don't have the tools to try to answer a question like this. If it helps I can say how the question arises: If the answer is no, as I suspect, then any non-trivial covering map $f:\mathbb D\to X$ is infinitely sheeted - this would say that holomorphicity is irrelevant to something that came up yesterday.)

REPLY [2 votes]: This is a special case of the Brouwer plane translation theorem. 
See the paper of John Franks entitled "A new proof of the Brouwer plane translation theorem", Ergodic Theory Dynam. Systems 12 (1992), no. 2, 217–226.