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Consider the space of orientation preserving diffeomorphism on the circle denoted by $\DeclareMathOperator{\Diff}{{\it Diff}}\Diff_+(S^1)$. Let $\frac{p}{q} \in \mathbb{Q}$. Im trying to prove that the subset of diffeomorphisms with rotation number $\frac{p}{q}$ is path connected.

i.e that the set $S := \{ f \in  \Diff_+(S^1) : \rho(f) = \frac{p}{q}\}$. Is path connected.

Let $f$, $g \in S$. Im considering an homotopy of the type

\begin{equation} H(t,x) = (1-t)F(x) + tG(x) \end{equation}

Where $F$ and $G$ are "the closest" lifts of $f, g$. Let us denote by $h$ the projection of $H$. It is obvious that $h$ is a diffeomorphism and orientation preserving since $H(t,\cdot)' > 0$ for all $t$. Now I have to prove that $H(t, \cdot)$ projects in $S$.

To do that i'm trying to see if $h$ has any periodic point of period $q$. The ordening of the orbit will be preserved so the rotation number is $\frac{p}{q}$.

I tried proving the existence of an $x$ such that $H^q(x) = x + p$, without much success. Also i tried to bound $\rho (H)$ by $\rho(F)$ and $\rho(G)$.

Any hints on how to proceed?

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