Displacement sequence of an orientation preserving circle homeomorphism

We give a complete description of the behaviour of the sequence of displacements ${\eta }_n(z)={\mathrm{\Phi }}^n(x)-{\mathrm{\Phi }}^{n-1}(x)\mathrm{o}d1$, $z=\exp(2\pi \rmi x)$,alongatrajectory${\varphi^{n}(z)}$,where$\varphi$ is an orientation preserving circle homeomorphism and $\Phi:$\mathbb{R}$ \to $\mathbb{R}$$itslift\mathrm{.}Iftherotationnumber$\varrho(\varphi)=\frac{p}{q}$ is rational then ${\eta }_n(z)$ is asymptotically periodic with semi-period $q$. This convergence to a periodic sequence is uniform in $z$ if we admit that some points are iterated backward instead of taking only forward iterations for all $z$. If $\varrho (\phi )\mathrm{\notin }\mathbb{Q}$ then the values of ${\eta }_n(z)$ are dense in a set which depends on the map $\gamma$ (semi-)conjugating $\phi$ with the rotation by $\varrho (\phi )$ and which is the support of the displacements distribution. We provide an effective formula for the displacement distribution if $\phi$ is $C^1$-diffeomorphism and show approximation of the displacement distribution by sample displacements measured along a trajectory of any other circle homeomorphism which is sufficiently close to the initial homeomorphism $\phi$. Finally, we prove that even for the irrational rotation number $\varrho$ the displacement sequence exhibits some regularity properties.