Displacement sequence of an orientation preserving circle homeomorphism

TL;DR

This paper thoroughly analyzes the displacement sequences of orientation-preserving circle homeomorphisms, revealing their asymptotic behavior, distribution, and regularity, with distinctions between rational and irrational rotation numbers.

Contribution

It provides a complete description of displacement sequences, formulas for their distribution, and shows how they can be approximated and analyzed for both rational and irrational rotation numbers.

Findings

Displacement sequences are asymptotically periodic for rational rotation numbers.

Displacement values are dense in a set depending on the semi-conjugating map for irrational rotation numbers.

Displacement distribution can be effectively approximated and exhibits regularity even for irrational rotations.

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.