Edit distance exponents for irrational rotations

Abstract

We study quantitative edit-distance asymptotics for symbolic codings of irrational rotations $x \mapsto x+\alpha$ on $\mathbb{T}$ in terms of the irrationality exponent $\mu(\alpha)$, the supremum of $\mu \in \mathbb{R}$ for which the inequality $0 < |\alpha - p/q| < q^{-\mu}$ has infinitely many solutions. For the binary coding determined by an interval $[0,\beta)$, let $\mathcal{W}_N$ be the set of length-$N$ words arising from all initial points $x$ under $x \mapsto x+\alpha$. We develop new techniques for estimating edit distance and compute the growth exponents of the edit-distance diameter $\mathrm{diam}_E(\mathcal{W}_N)$. For every $\alpha \notin \mathbb{Q}$ and almost every $\beta \in (0,1)$, we show that $\displaystyle (*) \quad \limsup_{N\to\infty}\frac{\log \mathrm{diam}_E(\mathcal{W}_N)}{\log N} = \frac{\mu(\alpha)-1}{\mu(\alpha)},$ and the corresponding $\liminf$ equals $1/2$. When $\mu(\alpha)-1$ is at most the golden mean $\varphi$, the asymptotics $(*)$ hold for all $\beta$. However, for $\mu>1+\varphi$, there is an uncountable set of $\alpha$ with $\mu(\alpha)=\mu$ for which the edit-distance exponents are strictly smaller than $(*)$ for uncountably many $\beta$. We also derive consequences for aperiodic circle homeomorphisms and Sturmian sequences. For rotations of $\mathbb{T}^d$ coded by boxes, we prove that for almost every rotation vector, the common edit-distance exponent is $d/(d+1)$. Finally, we raise the question of estimating edit-distance exponents for more general dynamical systems.