Dumont--Thomas numeration systems for $\mathbb{Z}$

We extend the well-known Dumont--Thomas numeration system to $\mathbb{Z}$ by considering two-sided periodic points of a substitution, thus allowing us to represent any integer in $\mathbb{Z}$ by a finite word (starting with $\mathtt{0}$ when nonnegative and with $\mathtt{1}$ when negative). We show that an automaton returns the letter at position $n\in\mathbb{Z}$ of the periodic point when fed with the representation of $n$. The numeration system naturally extends to $\mathbb{Z}^d$. We give an equivalent characterization of the numeration system in terms of a total order on a regular language. Lastly, using particular periodic points, we recover the well-known two's complement numeration system and the Fibonacci analogue of the two's complement numeration system. ; Comment: v1: 14 pages, 1 figure, 1 table. v2: 16 pages, added a section on a characterization of the numeration system by a total order on a regular language