Combinatorial substitutions and sofic tilings

A combinatorial substitution is a map over tilings which allows to define sets of tilings with a strong hierarchical structure. In this paper, we show that such sets of tilings are sofic, that is, can be enforced by finitely many local constraints. This extends some similar previous results (Mozes'90, Goodman-Strauss'98) in a much shorter presentation.Comment: 17 pages, 16 figures. In proceedings of JAC 201

Construction of the discrete hull for the combinatorics of a regular pentagonal tiling of the plane

The article 'A "regular" pentagonal tiling of the plane' by P. L. Bowers and K. Stephenson defines a conformal pentagonal tiling. This is a tiling of the plane with remarkable combinatorial and geometric properties. However, it doesn't have finite local complexity in any usual sense, and therefore we cannot study it with the usual tiling theory. The appeal of the tiling is that all the tiles are conformally regular pentagons. But conformal maps are not allowable under finite local complexity. On the other hand, the tiling can be described completely by its combinatorial data, which rather automatically has finite local complexity. In this paper we give a construction of the discrete hull just from the combinatorial data. The main result of this paper is that the discrete hull is a Cantor space

Algorithmic Complexity for the Realization of an Effective Subshift By a Sofic

Realization of d-dimensional effective subshifts as projective sub-actions of d + d\u27-dimensional sofic subshifts for d\u27 >= 1 is now well known [Hochman, 2009; Durand/Romashchenko/Shen, 2012; Aubrun/Sablik, 2013]. In this paper we are interested in qualitative aspects of this realization. We introduce a new topological conjugacy invariant for effective subshifts, the speed of convergence, in view to exhibit algorithmic properties of these subshifts in contrast to the usual framework that focuses on undecidable properties

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