Combinatorics of Pisot Substitutions

Siirretty Doriast

Decidability Problems for Self-induced Systems Generated by a Substitution

International audienceIn 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 tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties yields by the combinatorial substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems

Consistency of multidimensional combinatorial substitutions

Multidimensional combinatorial substitutions are rules that replace symbols by finite patterns of symbols in $\mathbb Z^d$. 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: it can fail to be consistent, and the patterns in an image by the substitution might overlap. We prove that it is undecidable whether a two-dimensional substitution is consistent or overlapping, and we provide practical algorithms to decide these properties in some particular cases.Comment: 13 pages, v2 includes corrections to match the published versio

Connectedness of fractals associated with Arnoux-Rauzy substitutions

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 sets, where each set is a renormalized projection of a finite union of faces of unit cubes. We exploit this combinatorial definition to prove the connectedness of the Rauzy fractal associated with any finite product of three-letter Arnoux-Rauzy substitutions.Comment: 15 pages, v2 includes minor corrections to match the published versio

Towards long-term standardised carbon and greenhouse gas observations for monitoring Europe's terrestrial ecosystems : a review

Research infrastructures play a key role in launching a new generation of integrated long-term, geographically distributed observation programmes designed to monitor climate change, better understand its impacts on global ecosystems, and evaluate possible mitigation and adaptation strategies. The pan-European Integrated Carbon Observation System combines carbon and greenhouse gas (GHG; CO2, CH4, N2O, H2O) observations within the atmosphere, terrestrial ecosystems and oceans. High-precision measurements are obtained using standardised methodologies, are centrally processed and openly available in a traceable and verifiable fashion in combination with detailed metadata. The Integrated Carbon Observation System ecosystem station network aims to sample climate and land-cover variability across Europe. In addition to GHG flux measurements, a large set of complementary data (including management practices, vegetation and soil characteristics) is collected to support the interpretation, spatial upscaling and modelling of observed ecosystem carbon and GHG dynamics. The applied sampling design was developed and formulated in protocols by the scientific community, representing a trade-off between an ideal dataset and practical feasibility. The use of open-access, high-quality and multi-level data products by different user communities is crucial for the Integrated Carbon Observation System in order to achieve its scientific potential and societal value.Peer reviewe

SUBSTITUTIVE ARNOUX-RAUZY SEQUENCES HAVE PURE DISCRETE SPECTRUM

uniform distribution theor

Topological substitution for the aperiodic Rauzy fractal tiling

27 pages, 13 figures. arXiv admin note: text overlap with arXiv:1101.3905International audienceWe consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a topological substitution (an object of purely combinatorial nature). We establish a link between the two families in a specific case, by defining an explicit topological substitution and by proving that it generates the same tilings as those associated with the Tribonacci Rauzy fractal

Connectedness of fractals associated with Arnoux-Rauzy substitutions

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 connectedness of the fractal associated with any finite product of Arnoux-Rauzy substitutions.