Brun expansions of stepped surfaces |
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Authors: | Valérie Berthé Thomas Fernique |
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Institution: | aLIAFA–Univ. Paris Diderot–Paris 7 & CNRS–Case 7014, 75205 Paris Cedex 13, France;bLIF–Univ. de Provence & CNRS–39 rue Joliot-Curie, 13453 Marseille, France |
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Abstract: | Dual maps have been introduced as a generalization to higher dimensions of word substitutions and free group morphisms. In this paper, we study the action of these dual maps on particular discrete planes and surfaces, namely stepped planes and stepped surfaces. We show that dual maps can be seen as discretizations of toral automorphisms. We then provide a connection between stepped planes and the Brun multi-dimensional continued fraction algorithm, based on a desubstitution process defined on local geometric configurations of stepped planes. By extending this connection to stepped surfaces, we obtain an effective characterization of stepped planes (more exactly, stepped quasi-planes) among stepped surfaces. |
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Keywords: | Arithmetic discrete plane Brun algorithm Digital planarity Discrete geometry Dual map Flip Free group morphism Multi-dimensional continued fraction Stepped plane Stepped surface Substitution |
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