Displacement sequence of an orientation preserving circle homeomorphism

Abstract

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