fact an actual conjugacy. In other words $\Phi_{F}:\R\to \R$ is an increasing homeomorphism.

If $\rho(f)=\alpha \in \R\setminus\Q$ and if $f$ is a ${\cal C}^2$ diffeomorphism, then $f:\T\to \T$ is conjugate to the rotation by angle $\alpha$.

The following two are basic results which can be found in any standard text in Dynamical Systems ([@katok-hasselblatt], see [@thesis] for \[prop\_transincrease\]). Let $\mathcal{D}^+(\mathbb R)$ be the set of increasing homeomorphisms of the real line which induce orientation presetving circle homeomorphisms. For $F, G \in \mathcal{D}^+(\mathbb R)$, we say $F>G$ if $F(x)>G(x)$ for all $x\in \R$.

\[lemma\_transincrease\] Assume that $F,G \in \mathcal{D}^+(\mathbb R)$. If $F>G$, then $\trans(F) \ge \trans(G)$, and this inequality is strict if one of $\trans(F)$ and $\trans(G)$ is irrational.

\[prop\_transincrease\] Assume that $F\in \cal D^+(\R)$ and $(F_t)_{t\in {\mathbb R}}$ be the one-parameter family of homeomorphisms of the real line defined as $$F_t(x):=F(x)+t.$$ Then $t\mapsto \trans(F_t)$ is continuous, non-decreasing and $$\trans(F_{t+1})=\trans(F_t)+1.$$ For all $\alpha \in {\mathbb R}, \trans^{-1}(\alpha)=I_{\alpha}$ is a closed interval. If $\alpha \in {\mathbb R \setminus \mathbb Q}, I_{\alpha}$ is a point. And for a rational $p/q$, $I_{p/q}$ is reduced to a point $t_0 \iff F_{t_0}^{\circ q}(x)-x \equiv p$.

Boundary points of $\cal T_{p/q}$ and parabolic fixed points
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In 