Displacement sequence of an orientation preserving circle homeomorphism

We give a complete description of the behaviour of the sequence of displacements $η_n(z)=Φ^n(x) - Φ^{n-1}(x) \ \rmod \ 1$, $z=\exp(2π\rmi x)$, along a trajectory $\{φ^{n}(z)\}$, where $φ$ is an orientation preserving circle homeomorphism and $Φ:\mathbb{R} \to \mathbb{R}$ its lift. If the rotation number $\varrho(φ)=\frac{p}{q}$ is rational then $η_n(z)$ 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 $\varrho(φ) \notin \mathbb{Q}$ then the values of $η_n(z)$ are dense in a set which depends on the map $γ$ (semi-)conjugating $φ$ with the rotation by $\varrho(φ)$ and which is the support of the displacements distribution. We provide an effective formula for the displacement distribution if $φ$ 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 $φ$. Finally, we prove that even for the irrational rotation number $\varrho$ the displacement sequence exhibits some regularity properties.

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