Brun expansions of stepped surfaces

TL;DR

This paper explores the action of dual maps on stepped surfaces, linking them to toral automorphisms and Brun continued fractions, providing new characterizations of stepped planes and quasi-planes.

Contribution

It introduces a novel connection between dual maps, stepped surfaces, and Brun continued fractions, extending the understanding of discretized toral automorphisms.

Findings

Dual maps act as discretizations of toral automorphisms.

A connection between stepped planes and Brun continued fractions is established.

Characterization of stepped quasi-planes among stepped surfaces is achieved.

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.