Itineraries of rigid rotations and diffeomorphisms of the circle

Details Itineraries of rigid rotations and diffeomorphisms of the circle Itineraries of rigid rotations and diffeomorphisms of the circle

2008-01-10

arxiv.org/abs/0801.1639v2

Mathematics

Dynamical Systems

Added error estimates in response to referees' comments

Scientific paper

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.

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