Attractors of sequences coding $\beta$-integers

In this paper, we describe minimal string attractors of prefixes of simple Parry sequences.

These sequences form a coding of distances between consecutive $\beta$-integers in numeration systems with a real base $\beta$.

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 minimal attractors of prefixes of some particular form of binary non-simple Parry sequences.