of $F$. As $G$ is analytic, either it has a double zero, or it vanishes at least once with positive derivative and once with negative derivative. This shows that counting multiplicities, $f$ has at least $2$ cycles with rotation number $p/q$.

Next we address the case of irrational rotation number.

\[prop\_semiconjugacy\] If $\rho(f) = \alpha \in \R \setminus\Q$, then $f:\T \to \T$ is semi-conjugate to the rotation $\T \ni [x]\mapsto [x+\alpha] \in
\T$.

In fact, the semiconjugacy may be obtained as follows. The sequence of maps $\Phi_N$ defined as $$\Phi_N(x)\eqdef \frac{1}{N}\sum_{k=0}^{N-1} \left(F^{\circ k}(x)-F^{\circ k}(0)\right)$$ converges, as $N\to +\infty$, to a non-decreasing continuous surjective map $\Phi_{F}:\R\to \R$, which satisfies $$\Phi_{F}(x+1) = \Phi_{F}(x)+1\quad\text{and}\quad \Phi_{F}\circ F(x) =
T_\alpha \circ \Phi_{F}(x)$$ where $T_\alpha: x\mapsto x+\alpha$ is the translation by $\alpha$.

The statement of semiconjugacy does not appear in Poincaré’s mémoire [@poincare] but it is equivalent to the following. In [@poincare], he proved that if $\rho(f)=\alpha$ is irrational, the order of points in the orbit of $f$ on the circle is the same as the order of points in the orbit of rotation by $\alpha$ on $\T$. The following result of Denjoy implies that when $F$ is an analytic diffeomorphism, then the semiconjugacy is in 