Diffeomorphism with same rotation number is path connected

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

$$$$H(t,x) = (1-t)F(x) + tG(x)$$$$

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?