I'm looking at the following problem from Katok & Hasselblatt (Modern Theory of Dynamical Systems, Problem $11.2.4$ if anyone cares):
Let $f:S^1\to S^1$ be an orientation-reversing homeomorphism of the circle. Show that $f$ has exactly two fixed points, and the rotation number of $f$ is zero.
Now, to start off with I use an easy consequence of the Lefschetz fixed point theorem, which says $f:S^n\to S^n$ has a fixed point if $\deg f\neq(-1)^{n+1}$. Since in our case, $\deg f=-1$, this certainly applies, so $f$ has at least one fixed point. Also, any map with a fixed point has zero rotation number, so $\tau(f)=0$ immediately. However, I'm stuck trying to show the existence of a second fixed point.
Can anybody throw a hint my direction?
Best Answer
Much much simpler, no need for Lefschtz.
Consider a lift $F$. Notice that $F(0)-0=(F(1)-1)+2$ (make the computations, it is orientation reversing). So, the $2$ gives you the answer!
Presumably you mean that $f^2$ has zero rotation number (makes no sense otherwise according to the definitions, in particular in KH). But indeed it follows as you say.