Derivatives of rotation number of one parameter families of circle diffeomorphisms (Q418264)

For \(t\in [-\frac{1}{2}, \frac{1}{2}]\) and an orientation-preserving \(C^{\infty }\)-diffeomorphism \(f\) of the circle set \(f_t :=R_t \circ f\), where \(R_t\) is the rotation by \(t\). The author shows that if the rotation number \(\rho (t)\) of \(f_t\) is irrational, then NEWLINE\[NEWLINE \limsup _{s\to t} \frac{\rho (s)-\rho (t)}{s-t}\geq 1. NEWLINE\]NEWLINE (He also remarks that this inequality is not valid for \(\liminf\).) To this end, a weaker version of the above result is first proved. (This version allows the author to give a new proof of a result of \textit{J. Graczyk} in [Bol. Soc. Bras. Mat., Nova Sér. 24, No. 2, 201--210 (1993; Zbl 0798.58048)].) Next, with the use of the Denjoy distortion lemma, the case when \(\rho (t)\) is a Liouville number is considered. Finally, the case of non-Liouville \(\rho (t)\) is studied. To do this, a result of \textit{P. Brunovský} from [Czech. Math. J. 24(99), 74--90 (1974; Zbl 0308.58007)] is applied.