Itineraries of rigid rotations and diffeomorphisms of the circle

Abstract

We examine the itinerary of $0\in S^1=\mathbb{R}/\mathbb{Z}$ under the rotation by $α\in\R\bs\Q$. 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.

Added error estimates in response to referees' comments