Nicolas Bédaride

No Weak Local Rules for the 4p-Fold Tilings

On the one hand, Socolar showed in 1990 that the n-fold planar tilings admit weak local rules when n is not divisible by 4 (the

Regular simplices and periodic billiard orbits

n=10

Symbolic Dynamics of a Piecewise Rotation: Case of the Non Symmetric Bijective Maps

n=10 case corresponds to the Penrose tilings and is known since 1974). On the other hand, Burkov showed in 1988 that the eightfold tilings do not admit weak local rules, and Le showed the same for the 12-fold tilings (unpublished). We here show that this is actually the case for all the 4p-fold tilings.

Thermodynamic formalism and Substitutions

We provide an effective characterization of the planar octagonal tilings which admit weak local rules. As a corollary, we show that they are all based on quadratic irrationalities, as conjectured by Thang Le in the 1990s.

Invariant measures for train track towers

A simplex is the convex hull of n + 1 points in R n which form an affine basis. A regular simplex ∆ n is a simplex with sides of the same length. We consider the billiard flow inside a regular simplex of R n. We show the existence of two types of periodic trajectories. One has period n + 1 and hits once each face. The other one has period 2n and hits n times one of the faces while hitting once any other face. In both cases we determine the exact coordinates for the points where the trajectory hits the boundary of the simplex.

Periodic Billiard Trajectories in Polyhedra

This paper introduces two tiles whose tilings form a one-parameter family of tilings which can all be seen as digitization of two-dimensional planes in the four-dimensional Euclidean space. This family contains the Ammann-Beenker tilings as the solution of a simple optimization problem.

An example of PET. Computation of the Hausdorff dimension of the aperiodic set

We consider a specific piecewise rotation of the plane that is continuous on two half-planes, as studied by some authors like Boshernitzan, Goetz and Quas. If the angle belongs to the set