A quantitative shrinking target result on Sturmian sequences for rotations

Jon Chaika; David Constantine

doi:10.3934/dcds.2018229

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2018, Volume 38, Issue 10: 5189-5204. Doi: 10.3934/dcds.2018229

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A quantitative shrinking target result on Sturmian sequences for rotations

Jon Chaika^1, and
David Constantine^2,

1.
Department of Mathematics, University of Utah, 155 S 1400 E, Room 233, Salt Lake City, UT 84112, USA
2.
Department of Mathematics and Computer Science, Wesleyan University, 265 Church Street, Middletown, CT 06459, USA

Received: January 31, 2018

Revised: April 30, 2018

Published: September 2018

The first author is supported by NSF grants DMS-1004372, 135500, 1452762, the Sloan Foundation, a Warnock chair, and a Poincaré chair.

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Abstract

Let $ R_α$ be an irrational rotation of the circle, and code the orbit of any point $ x$ by whether $ R_α^i(x) $ belongs to $ [0,α)$ or $ [α, 1)$ - this produces a Sturmian sequence. A point is undetermined at step $ j$ if its coding up to time $ j$ does not determine its coding at time $ j+1$. We prove a pair of results on the asymptotic frequency of a point being undetermined, for full measure sets of $ α$ and $ x$.

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Mathematics Subject Classification: Primary: 37E10, 37A05, 37B10.

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References

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