I would like to know what is understood by $f:\mathbb{S}^1\rightarrow\mathbb{S}^1$ ($\mathbb{S}^1$ is the unit circle) being a diffeomorphism, in this context.
Does differentiability on $\mathbb{S}^1$ mean that there exists an open set $U$ in $\mathbb{R}^2$ with $\mathbb{S}^1\subseteq U$ and $f$ differentiable on $U$?
If we consider a lift $F:\mathbb{R}\rightarrow\mathbb{R}$ of $f$, is it true that $F$ is a diffeomorphism if and only if $f$ is a diffeomorphism?
Differentiability means (or is equivalent to, since there are various possible definitions, all equivalent) differentiability of some lift (and so of all lifts).