I am studying involutions on $S^1$; namely, continous functions $f:S^1 \rightarrow S^1$ satisfying $f\circ f = id$. Examples of such maps are reflections along a given diameter, or map induced by extending rays from the boundary to a chosen point in $int(S^1)$. Using an algebraic topology argument, can be shown that $deg(f) = \pm 1$; that is, $f$ is either orientation-preserving or orientation-reversing.
I have read (without a convicent proof) that any involution is either free o has exactly two fixed points.
My concerns are explicitly the following
Is there any reference where I can find a proof of the above fact. I found the paper More on Involutions of a Circle by W:F Pfeffer, but I didn't get the proof.
Are the mentioned example the only involutions over $S^1$? is there any proof of this fact out there?
I appreciate any direction
Proving that involutions have 0 or 2 fixed points is not hard. Suppose that $\tau$ is an involution fixing a point $z$ in $S^1$. I identify $S^1$ with 1-point compactification of ${\mathbb R}$, so that $z=\infty$. Then, by restriction, $\tau$ defines a homeomorphism $f: {\mathbb R}\to {\mathbb R}$, hence, a monotonic function. If $f(x)> x$ for all $x\in {\mathbb R}$, then $$ f(f(x))> f(x)>x $$ and, hence, $f\circ f\ne id$; a contradiction. Similarly, if $f(x)< x$ for all $x$. Thus, the graph of $f$ has cross the line $y=x$, i.e. $f$ has to fix a point in ${\mathbb R}$. Now, let $A\subset {\mathbb R}$ denote the fixed point set of $f$; $A\ne {\mathbb R}$ is a closed subset. The map $f$ either preserves each component of $A^c={\mathbb R}- A$ or there are two complementary components and $f$ swaps them. In the latter case, $\tau$ has exactly two fixed points in $S^1$.
Suppose, therefore, that $f$ preserves each component of $A^c$. Every component is an open interval $I$ and the same argument proving that the involution $f: {\mathbb R}\to {\mathbb R}$ fixes a point in ${\mathbb R}$, shows that $f$ fixes a point in $I$, which is a contradiction.
Thus, $\tau$ has either no fixed points in $S^1$ or two. Working a bit more with this argument shows that $\tau$ is conjugate to a rotation or a reflection.