Itineraries of rigid rotations and diffeomorphisms of the circle

David Richeson; Paul Winkler; Jim Wiseman

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arXiv · 0801.1639

Itineraries of rigid rotations and diffeomorphisms of the circle

David Richeson·Paul Winkler·Jim Wiseman

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Abstract

We examine the itinerary of $0\in S^{1}=\R/\Z$ 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 $α$ 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 $g:S^{1}\to S^{1}$ 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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David Richeson, Paul Winkler, Jim Wiseman. 2009-09-30. Itineraries of rigid rotations and diffeomorphisms of the circle. https://arxiv.org/abs/0801.1639

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