Local rule substitutions and stepped surfaces

Abstract

Substitutions on words, i.e., non-erasing morphisms of the free monoid, are simple combinatorial objects which produce infinite words by iteratively replacing letters by words. This paper introduces a notion of substitution acting on multi-dimensional words, namely local rule substitutions. Roughly speaking, local rules play for multi-dimensional words the role played by the concatenation product for substitutions on words. We then particularly focus on the local rule substitutions which act on the two-dimensional words coding stepped surfaces, and we show that a wide class of them can be derived from generalized substitutions.