Tilings of the plane and Thurston semi-norm

Jean-René Chazottes (1) , Jean-Marc Gambaudo (2) , François Gautero (3)

1 CPHT - Centre de Physique Théorique [Palaiseau]
2 INLN - Institut Non Linéaire de Nice Sophia-Antipolis
3 JAD - Laboratoire Jean Alexandre Dieudonné

We show that the problem of tiling the Euclidean plane with a finite set of polygons (up to translation) boils down to prove the existence of zeros of a non-negative convex function defined on a fi nite-dimensional simplex. This function is a generalisation, in the framework of branched surfaces, of the Thurston semi-norm originally defi ned for compact 3-manifolds.

Mots clés: translation surfaces; Wang tilings; branched surfaces; translation surfaces.

Domaines: Systèmes dynamiques [math.DS]; Topologie géométrique [math.GT]

Jean-René Chazottes, Jean-Marc Gambaudo, François Gautero. Tilings of the plane and Thurston semi-norm. Geometriae Dedicata, 2014, 173, pp.129-142. ⟨hal-00769022v2⟩