the quantity $\ds \frac{F^{\circ n}(x)-x}{n}$. Poincaré showed that this quantity has a limit as $n\to \infty$ and the limit does not depend on the choice of the point $x$. We will call this limit as the of $F$ and denote this as $\trans(F)$. The translation number modulo 1 will be defined as the $\rho(f)$. This definition of translation and rotation numbers agrees with the fact that the translation number of a translation $T_\theta : x\mapsto x+\theta$ is $\theta$ and the rotation number of a rotation $R_\theta : [x]\mapsto [x+\theta]$ is equal to $\theta$ mod 1.

For a fixed $t$, the translation $x\mapsto x+t$ on the real line induces a circle homeomorphism which is equivalent to a rotation. One can perturb this translation with a non-linear factor and also can consider the map $$F_{t,a}: x \mapsto x+t+a\phi(x)$$ where $\phi$ is a Lipschitz continuous function of period $1$. There are two constants $a_{\min}$ and $a_{\max}$ which depend on $\phi$ such that if $t\in \R $ and $a_{\min}<a<a_{\max}$ then $F_{t,a}$ is monotone increasing homeomorphism and thus $F_{t,a}$ induces a circle homeomorphism. This is the of this family, denoted as $$\cal P \eqdef \{(t,a)~|~t \in \R, a_{\min}<a<a_{\max}\}.$$ Note that for $(t,a)\in \cal P$, the 