Reflection on the reflection complexity
Lubomíra Dvořáková, Edita Pelantová
TL;DR
The paper proves Allouche et al.'s conjecture that a sequence $oldsymbol{u}$ is eventually periodic iff there exists $n$ with $r_{oldsymbol{u}}(n+2)=r_{oldsymbol{u}}(n)$, grounding the result in a refined analysis of reflection classes and the quantity $ ext{T}(t)$. It then fully characterizes aperiodic sequences for which $r_{oldsymbol{u}}(n+2)=r_{oldsymbol{u}}(n)+1$ for all $n$, showing these are exactly Sturmian sequences or simple morphic images thereof via $oldsymbol{}: a o ac,\, b o bc$ with a new symbol $c$. For the larger-n regime, the authors show the equality holds for all sufficiently large $n$ precisely when the sequence is quasi-Sturmian with a reversal-closed suffix language. The work links reflection complexity to classical Sturmian theory, offering a clear dichotomy between periodic, Sturmian, and quasi-Sturmian structures through a robust combinatorial framework built on $r_{oldsymbol{u}}$, $ ext{Ind}_{oldsymbol{u}}(n)$, and Rauzy-graph arguments.
Abstract
The factor complexity ${\mathcal C}_{\mathbf u}$ of a sequence ${\mathbf u} = u_0u_1u_2 \cdots$ over a finite alphabet counts the number of factors of length $n$ occurring in $\mathbf u$, i.e., ${\mathcal C}_{\mathbf u}(n) = \#{\mathcal L}_n(\mathbf u)$, where ${\mathcal L}_n({\mathbf u)}= \{u_iu_{i+1}\cdots u_{i+n-1}: i \in \mathbb N\}$. Two factors of ${\mathcal L}_n(\mathbf u)$ are said to be equivalent if one factor is the reversal of the other one. Recently, Allouche et al. introduced the reflection complexity $r_{\mathbf u}$ which counts the number of non-equivalent factors of $\mathcal{L}_n(\mathbf u)$. They formulated the following conjecture: a sequence $\mathbf u$ is eventually periodic if and only if $r_{\mathbf u}(n+2) = r_{\mathbf u}(n)$ for some $n \in \mathbb N$. Here we prove the conjecture and characterize the sequences for which $r_{\mathbf u}(n+2) = r_{\mathbf u}(n)+1$ for every $n \in \mathbb N$ and also the sequences for which the equality is satisfied for every sufficiently large $n \in \mathbb N$.