I'm reading about Denjoy's theorem right now and noticed a few different forms. If $f$ is a diffeomorphism of $\mathbb{S}^1$ with bounded derivative and irrational rotation number, it is sometimes assumed that $f$ has positive derivative. However several authors like Katok and Hasselblatt in "Introduction to the modern theory of dynamical systems" don't make that assumption, yet in the proof they use a lemma that's only valid if the derivative of the iterations $f'(f^k(x))$ doesn't vanish for any $k \in \mathbb{Z}$. And in that lemma they actually clarify that the dervative must not vanish on a certain intervall but in the proof of Denjoy's theorem it's not explained why that should be the case for an arbitrary $f$. Brin and Stuck do the same in "Introduction to Dynamical Systems", except they simply use $\log(f')$ without mentioning if or why $f'$ can't be zero.
Is it possible to assume for every diffeomorphism of the mentioned form that the derivative never vanishes?
Thanks!
I guess I found the answer. The derivative of the homeomorphism $f: \mathbb{S}^1 \to \mathbb{S}^1$ exists at a point $x\in \mathbb{S}^1$ iff the derivative of a lift $F:\mathbb{R} \to \mathbb{R}$ exists at every point $\pi^{-1}(x)$ where $\pi$ is the natural projection. The derivatives also must have the same value. However a homeomorphism of $\mathbb{R}$ has to be monotone and can't have derivative $0$ since it's inverse wouldn't be differentiable then. Consequently we can assume that the derivative of $f$ can't vanish.