Published as: Richeson, David, Paul Winkler, and Jim Wiseman. Itineraries of Rigid Rotations and Diffeomorphisms of the Circle. Theoretical Computer Science 411, no. 1 (2010): 259-265. https://www.sciencedirect.com/science/article/pii/S030439750900704X © 2009 Elsevier B.V. All rights reserved. This author post-print is made available on Dickinson Scholar with the permission of the publisher. For more information on the published version, visit Science Direct's Website. © 2010. This publication is made available under the CC-BY-NC-ND 4.0 license: http://creativecommons.org/licenses/by-nc-nd/4.0/

We examine the itinerary of 0 ∈ S1 = R/Z under the rotation by α ∈ R \ Q. The motivating question is: if we are given only the itinerary of 0 relative to I ⊂ S1 , a finite union of closed intervals, can we recover α and I? We prove that the itineraries do determine α and I up to certain equivalences. Then we present elementary methods for finding α and I. Moreover, if g : S1 → S1 is a C2 , orientation preserving diffeomorphism with an irrational rotation number, then we can use the orbit itinerary to recover the rotation number up to certain equivalences.