Edit distance exponents for irrational rotations

Abstract

We study quantitative edit-distance asymptotics for symbolic codings of irrational rotations $x \mapsto x+α$ on $\mathbb{T}$ in terms of the irrationality exponent $μ(α)$, the supremum of $μ\in \mathbb{R}$ for which the inequality $0 < |α- p/q| < q^{-μ}$ has infinitely many solutions. For the binary coding determined by an interval $[0,β)$, let $\mathcal{W}_N$ be the set of length-$N$ words arising from all initial points $x$ under $x \mapsto x+α$. 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 $α\notin \mathbb{Q}$ and almost every $β\in (0,1)$, we show that $\displaystyle (*) \quad \limsup_{N\to\infty}\frac{\log \mathrm{diam}_E(\mathcal{W}_N)}{\log N} = \frac{μ(α)-1}{μ(α)},$ and the corresponding $\liminf$ equals $1/2$. When $μ(α)-1$ is at most the golden mean $φ$, the asymptotics $(*)$ hold for all $β$. However, for $μ>1+φ$, there is an uncountable set of $α$ with $μ(α)=μ$ for which the edit-distance exponents are strictly smaller than $(*)$ for uncountably many $β$. 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.