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

We extend the well-known Dumont–Thomas numeration systems to $\mathbb{Z}$ using an approach inspired by the two’s complement numeration system. Integers in $\mathbb{Z}$ are canonically represented by a finite word (starting with $\mathtt{0}$ when nonnegative and with $\mathtt{1}$ when negative). The systems are based on two-sided periodic points of substitutions as opposed to the right-sided fixed points. For every periodic point of a substitution, we construct an automaton which returns the letter at position $n\in \mathbb{Z}$ of the periodic point when fed with the representation of $n$ in the corresponding numeration system. 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.