Two-dimensional iterated morphisms and discrete planes

Authors:
Pierre Arnoux;Valérie Berthé;Anne Siegel
Affiliations:
IML UPR-CNRS 9016, Campus de Luminy case 907, 13288 Marseille Cedex 9, France;LIRMM UMR-CNRS 5506, 161 rue Ada, 34392 Montpellier Cedex 5, France;IRISA UMR-CNRS 6074, Campus de Beaulieu, 35042 Rennes Cedex, France
Venue:
Theoretical Computer Science - Combinatorics of the discrete plane and tilings
Year:
2004

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Abstract

Iterated morphisms of the free monoid are very simple combinatorial objects which produce infinite sequences by replacing iteratively letters by words. The aim of this paper is to introduce a formalism for a notion of two-dimensional morphisms; we show that they can be iterated by using local rules, and that they generate two-dimensional patterns related to discrete approximations of irrational planes with algebraic parameters. We associate such a two-dimensional morphism with any usual Pisot unimodular one-dimensional iterated morphism over a three-letter alphabet.