Itineraries of rigid rotations and diffeomorphisms of the circle

We examine the itinerary of 0 ∈ S = 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 ⊂ S, 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 : S → S is a C, orientation preserving diffeomorphism with an irrational rotation number, then we can use the orbit itinerary to recover the rotation number up to certain equivalences. A useful and common technique for analyzing discrete dynamical systems is to partition the space and study the itineraries and corresponding shift spaces of orbits. Often the dynamics of the original map is complicated (or chaotic) and the shift space provides a convenient way of extracting properties of the original dynamical system. In this paper, however, we consider orbit itineraries for one of the most elementary and well-understood dynamical systems—the rigid rotation of a circle. Let rα : S 1 = R/Z → S be the rotation rα(z) = z + α mod 1, where α ∈ R. Let I be a finite union of closed intervals (with nonempty interiors) in S that is neither ∅ nor S. The itinerary for 0 ∈ S is (a0, a1, a2, . . .), where