Billiard complexity in rational polyhedra

We give a new proof for the directional billiard complexity in the cube, which was conjectured in \cite{Ra} and proven in \cite{Ar.Ma.Sh.Ta}. Our technique gives us a similar theorem for some rational polyhedra.Comment: 9 pages, 4 figure

Outer billiard outside regular polygons

We consider outer billiard outside regular convex polygons. We deal with the case of regular polygons with $\{3,4,5,6,10\}$ sides, and we describe the symbolic dynamics of the map and compute the complexity of the language.Comment: 53 pages, 12 figure

Geometric realizations of two dimensional substitutive tilings

We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove that there is no primitive substitutive tiling of the hyperbolic plane $\mathbb{H}^2$. However, we give an example of substitutive tiling of \Hyp^2 which is non-primitive.Comment: 30 pages, 13 figure

CLASSIFICATION OF ROTATIONS ON THE TORUS T 2

International audienceWe consider a rotation on the torus T 2. We classify these rotations along their complexity functions. This can be seen as a generalization of Morse Hedlund theorem to the dimension two

DIRECTIONAL COMPLEXITY OF THE HYPERCUBIC BILLIARD

International audienceWe consider a minimal rotation on the torus T d of direction ω. A natural cellular decomposition of the torus is associated to this map. We consider an infinite orbit for this map. We compute the complexity of the associated word. Under some hypothesis on the direction, we obtain an exact formula which shows that the order of magnitude is n d. This result is related to the billiard map inside a hypercube of R d+1

Entropy of polyhedral billiard

International audienceWe consider the billiard map in a convex polyhedron of R 3 , and we prove that it is of zero topological entropy

PERIODIC BILLIARD TRAJECTORIES IN POLYHEDRA

International audienceWe consider the billiard map inside a polyhedron. We give a condition for the stability of the periodic trajectories. We apply this result to the case of the tetrahedron. We deduce the existence of an open set of tetrahedra which have a periodic orbit of length four (generalization of Fagnano's orbit for triangles), moreover we can study completly the orbit of points along this coding