Effective S-adic symbolic dynamical systems

Valérie Berthé (1) , Thomas Fernique (2) , Mathieu Sablik (3)

1 IRIF (UMR_8243) - Institut de Recherche en Informatique Fondamentale
2 LIPN - Laboratoire d'Informatique de Paris-Nord
3 I2M - Institut de Mathématiques de Marseille

We focus in this survey on effectiveness issues for S-adic subshifts and tilings. An S-adic subshift or tiling space is a dynamical system obtained by iterating an infinite composition of substitutions, where a substitution is a rule that replaces a letter by a word (that might be multi-dimensional), or a tile by a finite union of tiles. Several notions of effectiveness exist concerning S-adic subshifts and tiling spaces, such as the computability of the sequence of iterated substitutions, or the effectiveness of the language. We compare these notions and discuss effectiveness issues concerning classical properties of the associated subshifts and tiling spaces, such as the computability of shift-invariant measures and the existence of local rules (soficity). We also focus on planar tilings.

Mots clés: Symbolic dynamics; Adic map; Substitution; S-adic system; Planar tiling; Local rules; Sofic subshift; Subshift of finite type; Computable invariant measure; Effective language

Domaines: Systèmes dynamiques [math.DS]

Valérie Berthé, Thomas Fernique, Mathieu Sablik. Effective S-adic symbolic dynamical systems. 12th Conference on Computability in Europe, CiE 2016, Univ Paris 13; Univ Paris 7; Assoc Computabil Europe; Assoc Symbol Log; European Assoc Theoret Comp Sci, Jun 2016, Paris, France. pp.13-23, ⟨10.1007/978-3-319-40189-8_2⟩. ⟨hal-01488988⟩