Thomas Fernique

Functional stepped surfaces, flips, and generalized substitutions

A substitution is a non-erasing morphism of the free monoid. The notion of multidimensional substitution of non-constant length acting on multidimensional words is proved to be well-defined on the set of two-dimensional words related to discrete approximations of irrational planes. Such a multidimensional substitution can be associated with any usual unimodular substitution. The aim of this paper is to extend the domain of definition of such multidimensional substitutions to functional stepped surfaces. One central tool for this extension is the notion of flips acting on tilings by lozenges of the plane.

Generation and recognition of digital planes using multi-dimensional continued fractions

This paper extends, in a multi-dimensional framework, pattern recognition techniques for generation or recognition of digital lines. More precisely, we show how the connection between chain codes of digital lines and continued fractions can be generalized by a connection between tilings and multi-dimensional continued fractions. This leads to a new approach for generating and recognizing digital hyperplanes.

Brun expansions of stepped surfaces

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.

A characterization of flip-accessibility for rhombus tilings of the whole plane

It is known that any two rhombus tilings of a polygon are flip-accessible, that is, linked by a finite sequence of local transformations called flips. This paper considers flip-accessibility for rhombus tilings of the whole plane, asking whether any two of them are linked by a possibly infinite sequence of flips. The answer turning out to depend on tilings, a characterization of flip-accessibility is provided. This yields, for example, that any tiling by Penrose tiles is flip-accessible from a Penrose tiling.

Local Rules for Computable Planar Tilings

Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question is to characterize, among a class of non-periodic tilings, the aperiodic ones. In this paper, we answer this question for the well-studied class of non-periodic tilings obtained by digitizing irrational vector spaces. Namely, we prove that such tilings are aperiodic if and only if the digitized vector spaces are computable.

Local rule substitutions and stepped surfaces

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.

Stochastic flips on two-letter words

This paper introduces a simple Markov process inspired by the problem of quasicrystal growth. It acts over two-letter words by randomly performing flips, a local transformation which exchanges two consecutive different letters. More precisely, only the flips which do not increase the number of pairs of consecutive identical letters are allowed. Fixed-points of such a process thus perfectly alternate different letters. We show that the expected number of flips to converge towards a fixed-point is bounded by O(n3) in the worst-case and by O(n5/2 ln n) in the average-case, where n denotes the length of the initial word.

Bidimensional sturmian sequences and substitutions

Substitutions are powerful tools to study combinatorial properties of sequences. There exist strong characterizations through substitutions of the Sturmian sequences that are S-adic, substitutive or a fixed-point of a substitution. In this paper, we define a bidimensional version of Sturmian sequences and look for analogous characterizations. We prove in particular that a bidimensional Sturmian sequence is always S-adic and give sufficient conditions under which it is either substitutive or a fixed-point of a substitution.

No Weak Local Rules for the 4p-Fold Tilings

On the one hand, Socolar showed in 1990 that the n-fold planar tilings admit weak local rules when n is not divisible by 4 (the