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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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  • [1] V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel, editors, Substitutions in Dynamics, Arithmetics, and Combinatorics, volume 1794 of Lecture Notes in Mathematics. Springer, Berlin, 2002. doi: 10.1007/b13861.
    [2] J. Chaika and D. Constantine, Quantitative shrinking target properties for rotations and interval exchanges, To appear in Israel Journal of Mathematics, arXiv: 1201.0941.
    [3] H. Kesten, On a conjecture of Erdős and Szüz related to uniform distribution mod 1, Acta Arithmetica, 12 (1966), 193-212.  doi: 10.4064/aa-12-2-193-212.
    [4] A. Ya. Khinchin, Continued Fractions, Dover Books on Mathematics. Dover, 1997.
    [5] M. Lothaire, Algebraic Combinatorics on Words, volume 90 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511566097.
    [6] M. Morse and G. A. Hedlund, Symbolic dynamics Ⅱ. Sturmian trajectories, American Journal of Mathematics, 62 (1940), 1-42.  doi: 10.2307/2371431.
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