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A quantitative shrinking target result on Sturmian sequences for rotations

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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  • 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$.

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


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