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October  2018, 38(10): 5189-5204. doi: 10.3934/dcds.2018229

## A quantitative shrinking target result on Sturmian sequences for rotations

 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  February 2018 Revised  May 2018 Published  July 2018

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

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 $x$.

Citation: Jon Chaika, David Constantine. A quantitative shrinking target result on Sturmian sequences for rotations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5189-5204. doi: 10.3934/dcds.2018229
##### References:
 [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. Google Scholar [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.Google Scholar [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. Google Scholar [4] A. Ya. Khinchin, Continued Fractions, Dover Books on Mathematics. Dover, 1997. Google Scholar [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. Google Scholar [6] M. Morse and G. A. Hedlund, Symbolic dynamics Ⅱ. Sturmian trajectories, American Journal of Mathematics, 62 (1940), 1-42. doi: 10.2307/2371431. Google Scholar

show all references

##### References:
 [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. Google Scholar [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.Google Scholar [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. Google Scholar [4] A. Ya. Khinchin, Continued Fractions, Dover Books on Mathematics. Dover, 1997. Google Scholar [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. Google Scholar [6] M. Morse and G. A. Hedlund, Symbolic dynamics Ⅱ. Sturmian trajectories, American Journal of Mathematics, 62 (1940), 1-42. doi: 10.2307/2371431. Google Scholar
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