Lifts of orientation preserving diffeomorphisms of the circle

Question

I read a few discussions here such as

Orientation-preserving diffeomorphism

Orientation preserving diffeomorphism.

Structural stability in orientation preserving $[0,1]$ diffeomorphism

I am still confused about the following:

Consider orientation preserving diffeomorphisms of the circle, or equivalently, their lifts to the real line: $$\phi: \mathbf{R}^1 \rightarrow \mathbf{R}^1$$ and $$\phi(x)=x+\tilde{\eta}(x), \ \ \text{ with } \tilde{\eta}(x+1)=\tilde{\eta}(x) \ \ \text{ and } \tilde{\eta}'(x)>-1$$

Why does it say $$\tilde{\eta}'(x)>-1?$$

Thanks!

Answer 1

Since the orientation of the circle is induced by the covering $\mathbb R\to{\mathbb S}^1$, a diffeo of the circle preserves orientation iff its lifts do (note here that lifts are related by translation). The diffeo $\phi$ of $\mathbb R$ preserves orientation iff $\phi'(x)>0$, iff $\eta'(x)=(\phi(x)-x)'=\phi'(x)-1>0-1=-1$.