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Nicolas Bedaride

Publications and source records attributed to Nicolas Bedaride.

10 recordsLinked to original sources
arXiv2021

Cohomology groups for spaces of 12-fold tilings

We consider tilings of the plane with 12-fold symmetry obtained by the cut and projection method. We compute their cohomology groups using the techniques introduced by the second author, Hunton and Kellendonk. To do this we completely describe the window, the orbits of lines under the group action and the orbits of 0-singularities. The complete family of generalized 12-fold tilings can be described using 2-parameters and it presents a surprisingly rich cohomological structure. To put this finding into perspective, one should compare our results with the cohomology of the generalized 5-fold tilings (more commonly known as generalized Penrose tilings). In this case the tilings form a 1-parameter family, which fits in simply one of two types of cohomology.

math.KT↗
arXiv2019

Thermodynamic formalism and Substitutions

This paper studies properties of a Renormalization Operator for potentials in symbolic dynamics. These operators first appeared in \cite{BLL} and the link with substitutions was done in \cite{BL1}. Their fixed points are natural candidates to have pathologic behavior such as phase transitions. If $R$ is such an operator, we study the convergence of $R^{n}(φ)$ to the non-nul fixed point. We define the family of marked substitutions, which contains the Thue-Morse substitution, and show that the associated renormalization operators on potentials admits a unique non-nul continuous fixed point. Then, we show that $R^{n}(φ)$ converges to the fixed point as soon as $φ$ has the right germ close to $\mathbb K$.

math.DS↗
arXiv2012

Regular simplices and periodic billiard orbits

A simplex is the convex hull of $n+1$ points in $\mathbb{R}^{n}$ which form an affine basis. A regular simplex $Δ^n$ is a simplex with sides of the same length. We consider the billiard flow inside a regular simplex of $\mathbb{R}^n$. 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.

math.DS↗
arXiv2012

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.

math.GT↗
arXiv2012

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.

math.DS↗
arXiv2011

Billiard complexity in the hypercube

We consider the billiard map in the hypercube of $\mathbb{R}^d$. We obtain a language by coding the billiard map by the faces of the hypercube. We investigate the complexity function of this language. We prove that $n^{3d-3}$ is the order of magnitude of the complexity.

math.DS↗
arXiv2011

Periodic billiard trajectories in polyhedra

We 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.

math.DS↗
arXiv2011

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.

math.DS↗
arXiv2010

On the Number of Balanced Words of Given Length and Height over a Two-Letter Alphabet

We exhibit a recurrence on the number of discrete line segments joining two integer points in the plane using an encoding of such segments as balanced words of given length and height over the two-letter alphabet $\{0,1\}$. We give generating functions and study the asymptotic behaviour. As a particular case, we focus on the symmetrical discrete segments which are encoded by balanced palindromes.

math.CO↗