Itineraries of rigid rotations and diffeomorphisms of the circle

2008/01/10 by David Richeson, Richeson, David, Paul Winkler +3
Mathematics · #37B10 #37E10 #37E45 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:37B10 #msc:37E10 #msc:37E45

paper · pdf · doi:10.48550/arxiv.0801.1639

Added error estimates in response to referees' comments

arxiv created 2009/09/30 · arxiv updated 2009/12/01

Abstract

We examine the itinerary of 0∈ S¹=\R/\Z under the rotation by α∈\R\bs\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.

Related

• Rotation Numbers for S² diffeomorphisms
• Counting rotational subsets of the circle ℝ/ℤ under the angle multiplying map t↦ dt
• Pseudo-rotations of the closed annulus : variation on a theorem of J. Kwapisz
• Rotational subsets of the circle
• Rotation and periodicity in plane separating continua