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arXiv · 1608.03965

Brun expansions of stepped surfaces

Valérie Berthé·Thomas Fernique
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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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Valérie Berthé, Thomas Fernique. 2016-08-13. Brun expansions of stepped surfaces. https://doi.org/10.1016/j.disc.2010.12.007

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