Itineraries of rigid rotations and diffeomorphisms of the circle

TL;DR

This paper investigates how the itinerary of a point under irrational circle rotations can uniquely determine the rotation number and interval, providing methods for recovery and extending results to smooth diffeomorphisms.

Contribution

It proves that itineraries uniquely determine the rotation number and interval up to equivalence, and introduces elementary methods for their recovery, extending to smooth diffeomorphisms.

Findings

Itineraries determine the rotation number and interval up to equivalence.

Elementary methods are provided for recovering the rotation parameters.

The approach extends to $C^{2}$ smooth, orientation-preserving diffeomorphisms with irrational rotation number.

Abstract

We examine the itinerary of $0\in S^{1}=\R/\Z$ under the rotation by $\alpha\in\R\bs\Q$. The motivating question is: if we are given only the itinerary of 0 relative to $I\subset S^{1}$, a finite union of closed intervals, can we recover $\alpha$ and $I$? We prove that the itineraries do determine $\alpha$ and $I$ up to certain equivalences. Then we present elementary methods for finding $\alpha$ and $I$. Moreover, if $g:S^{1}\to S^{1}$ is a $C^{2}$, orientation preserving diffeomorphism with an irrational rotation number, then we can use the orbit itinerary to recover the rotation number up to certain equivalences.