Multidimensional generalized automatic sequences and shape-symmetric morphic words

An infinite word is S -automatic if, for all n ≥ 0 , its ( n + 1 ) th letter is the output of a deterministic automaton fed with the representation of n in the numeration system S . 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 ≥ 1 , we show that a multidimensional infinite word x : N d → Σ over a finite alphabet Σ is S -automatic for some abstract numeration system S built on a regular language containing the empty word if and only if x is the image by a coding of a shape-symmetric infinite word.