Circle homeomorphisms and rotation number

the map $z=e^{2 \pi i x} \mapsto \varphi (z)=e^{2 \pi i f(x)}$ defines a homeomorphism of the circle $S^{1}=\left\{ z \mid \left|z\right|=1\right\}$, preserving the cyclic order of any three points, and that an arbitrary orientation-preserving homeomorphism of the circle can be defined in this way;

if the limit $\lim_{n \rightarrow \infty } \frac{f^{n}(x)}{n}=\alpha$ exists for some $x_{0} \in \mathbb {R}$, then it also exists for every $x \in \mathbb {R}$;

if the map $\varphi^{m}$ has a fixed point for some $m \in \mathbb {N}$, then the limit $\alpha$ exists and is a rational number;

if no power of $\varphi$ has a fixed point, then the limit $\alpha$ exists and is an irrational number;

if two distinct functions $f_{1}$ and $f_{2}$ define the same homeomorphism $\varphi$, then the corresponding limits $\alpha_{1}$ and $\alpha_{2}$ differ by an integer.