Displacement sequence of an orientation preserving circle homeomorphism

Abstract

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 $\{{\phi }^n(z)\}$, where $\phi$ is an orientation preserving circle homeomorphism and $\Phi :\mathbb{R}\to \mathbb{R}$ its lift. If the rotation number $\varrho (\phi )=\frac{p}{q}$ 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 $\varrho (\phi )\notin \mathbb{Q}$ 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.

18 pages, 1 figure

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