A quantitative shrinking target result on Sturmian sequences for rotations

TL;DR

This paper investigates the frequency at which points in Sturmian sequences, generated by irrational rotations, remain undetermined over time, providing asymptotic results for typical parameters.

Contribution

It establishes new asymptotic bounds on the occurrence of undetermined points in Sturmian sequences for a broad set of rotation parameters.

Findings

Asymptotic frequency bounds for undetermined points

Results hold for full measure sets of  and initial points

Provides quantitative shrinking target results in symbolic dynamics

Abstract

Let $R_\alpha$ be an irrational rotation of the circle, and code the orbit of any point $x$ by whether $R_\alpha^i(x)$ belongs to $[0,\alpha)$ or $[\alpha,1)$ -- 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 $\alpha$ and $x$.