A quantitative shrinking target result on Sturmian sequences for rotations

Jon Chaika,David Constantine,

doi:10.3934/dcds.2018229

Abstract

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

Highlights

In this paper we study a shrinking target problem
Rather than being subject to some pre-determined analytic constraint the targets arise from the dynamics of the rotation itself
Note that the elements ai of the continued fraction depend on α; we will at times write ai(α) to emphasize this dependence

Summary

Statement of the problem and main results

In this paper we study a shrinking target problem. Let {Bi} be a sequence of measurable sets in [0, 1). By the Borel-Cantelli Lemma, the cases of real interest are when Several results on this problem for rotations and for interval exchange transformations are contained in [CC17], including the following. Sturmian sequence, circle rotation, symbolic dynamics, continued fractions. In this paper we consider another shrinking target problem for rotations, but one whose targets arise in a very different way. Rather than being subject to some pre-determined analytic constraint (as for the sequence {ri}) the targets arise from the dynamics of the rotation itself. To understand why Theorem A constitutes a shrinking target problem, consider the following. Theorem A is in some sense the best one can hope for in this setting, an interesting contrast with the stronger results obtained for targets of the form B(y, ri)

Notation and an outline of the paper

Failure of a stronger convergence

Proof of Theorem A