Attractors of sequences coding $β$-integers

TL;DR

This work analyzes minimal string attractors for prefixes of Parry sequences that code the distances between consecutive $β$-integers in non-standard numeration systems with base $β>1$. It proves that simple Parry sequences admit attractors of alphabet size for all prefixes, and under additional parametric conditions, these attractors are contained in the set $igl\{|φ^{n}(0)|-1\bigr\}$, with a second result relaxing conditions while preserving alphabet-size attractors. The authors also derive attractors for prefixes of a specific binary non-simple Parry sequence, illustrating the complexity added by non-simplicity. By connecting the morphic structure of Parry sequences to prefix attractors, the paper advances understanding of attractor-based descriptions in β-numeration and highlights open directions for more general fixed points of morphisms.

Abstract

In this paper, we describe minimal string attractors of prefixes of simple Parry sequences. These sequences form a coding of distances between consecutive $β$-integers in numeration systems with a real base $β$. Simple Parry sequences have been recently studied from this point of view and attractors of prefixes have been described. However, the authors themselves had doubts about their minimality and conjectured that attractors of alphabet size should be sufficient. We confirm their conjecture. Moreover, we provide attractors of prefixes of some particular form of binary non-simple Parry sequences.

Figures (1)

• Figure 1: Illustration of coding of distances in $\mathbb Z_\beta$ for $\beta=\frac{1+\sqrt{5}}{2}$

Theorems & Definitions (55)

• Definition 1
• Lemma 3
• Corollary 4
• Definition 5
• Example 6
• Lemma 7
• Lemma 8
• Definition 9
• Proposition 10: Parry condition
• Example 11