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- Valérie Berthé, Timo Jolivet, Anne Siegel
- ArXiv
- 2011

We prove that the symbolic dynamical system generated by a purely substitutive Arnoux-Rauzy sequence is measurably conjugate to a toral translation. The proof is based on an explicit construction of a fundamental domain with fractal boundary (a Rauzy fractal) for this toral translation. Communicated by Pierre Liardet Dedicated to the memory of Gérard Rauzy

- Valérie Berthé, Damien Jamet, Timo Jolivet, Xavier Provençal
- DGCI
- 2013

An arithmetical discrete plane is said to have critical connecting thickness if its thickness is equal to the infimum of the set of values that preserve its 2-connectedness. This infimum thickness can be computed thanks to the fully subtractive algorithm. This multidimensional continued fraction algorithm consists, in its linear form, in subtracting the… (More)

- Valérie Berthé, Jérémie Bourdon, Timo Jolivet, Anne Siegel
- WORDS
- 2013

Given a finite set S of unimodular Pisot substitutions, we provide a method for characterizing the infinite sequences over S that allow to generate a full discrete plane when, starting from a finite seed, we iterate the multidimensional dual substitutions associated with S. We apply our results to study the substitutions associated with the Brun… (More)

- Valérie Berthé, Jérémie Bourdon, Timo Jolivet, Anne Siegel
- ArXiv
- 2014

We define a generic algorithmic framework to prove pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and… (More)

- Timo Jolivet, Jarkko Kari
- Theor. Comput. Sci.
- 2012

Multidimensional combinatorial substitutions are rules that replace symbols by finite patterns of symbols in Zd. We focus on the case where the patterns are not necessarily rectangular, which requires a specific description of the way they are glued together in the image by a substitution. Two problems can arise when defining a substitution in such a way:… (More)

- Timo Jolivet, Jarkko Kari
- MFCS
- 2014

We study the decidability of some properties of self-affine sets specified by a graph-directed iterated function system (GIFS) with rational coefficients. We focus on topological properties and we prove that having empty interior is undecidable in dimension two. These results are obtained by studying a particular class of self-affine sets associated with… (More)

- Valérie Berthé, Timo Jolivet, Anne Siegel
- RAIRO - Theor. Inf. and Applic.
- 2014

Rauzy fractals are compact sets with fractal boundary that can be associated with any unimodular Pisot irreducible substitution. These fractals can be defined as the Hausdorff limit of a sequence of compact planar sets, where each set is the projection of a finite union of faces of unit cubes. We exploit this combinatorial definition to prove the… (More)

- Timo Jolivet, Anne Siegel
- MCU
- 2015

In this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on… (More)

Cellular automata are the most natural discretization of dynamical systems. Their long-term behaviour is captured in their limit set, also known as the maximal attractor. We investigate what complexity limit sets can have from the point of view of language theory, and we try to give some dynamical characterizations of subshifts that are limit sets. Then,… (More)

We prove that every free group of finite rank can be realized as the fundamental group of a planar Rauzy fractal associated with a 4-letter unimodular cubic Pisot substitution. This characterizes all countable fundamental groups for planar Rauzy fractals. We give an explicit construction relying on two operations on substitutions: symbolic splittings and… (More)