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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 and Continuous Dynamical Systems, 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.

[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.

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.

[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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