Multidimensional generalized automatic sequences and shape-symmetric morphic words

An infinite word is SS-automatic if, for all n≥0n≥0, its (n+1)(n+1)th letter is the output of a deterministic automaton fed with the representation of nn in the numeration system SS. In this paper, we consider an analogous definition in a multidimensional setting and study its relation to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for d≥1d≥1, we show that a multidimensional infinite word x:Nd→Σx:Nd→Σ over a finite alphabet ΣΣ is SS-automatic for some abstract numeration system SS built on a regular language containing the empty word if and only if xx is the image by a coding of a shape-symmetric infinite word.