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Nicolas Bédaride

Publications and source records attributed to Nicolas Bédaride.

At least 19 recordsLinked to original sources
arXiv2026

On balance properties of hypercubic billiard words

This paper studies balance properties for billiard words. Billiard words generalize Sturmian words by coding trajectories in hypercubic billiards. In the setting of aperiodic order, they also provide the simplest examples of quasicrystals, as tilings of the line obtained via cut and project sets with a cubical canonical window. By construction, the number of occurrences of each letter in a factor (i.e., a string of consecutive letters) of a hypercubic billiard word only depends on the length of the factor, up to an additive constant. In other words, the difference of the number of occurrences of each letter in factors of the same length is bounded. In contrast with the behaviour of letters, we prove the existence of words that are not balanced in billiard words: the difference of the number of occurrences of such unbalanced factors in longer factors of the same length is unbounded. The proof relies both on topological methods inspired by tiling cohomology and on arithmetic results on bounded remainder sets for toral translations.

math.DS↗
arXiv2025

Freezing phase transition for the Thue-Morse subshift

On the full shift on two symbols, we consider the potential defined by $V(x) = \frac{1}{n}$ where $n$ denotes the longest common prefix between the infinite word $x$ and an element of the subshift associated to the Thue-Morse substitution. Given a non negative real number $β$, the pressure function is $P(β):=\sup\left\{h_μ+β\int V\,dμ\right\},$ where the supremum is taken over all shift invariant probabilities $μ$ on the full shift and $h_μ$ is the Kolmogorov entropy. We prove that there is a freezing phase transition for the potential $V$: For $β$ large enough, the pressure $P(\be)$ is equal to zero. Similar results were previously published by Bruin and Leplaideur in \cite{BL2}, \cite{Bruin-Leplaid-13} but their proofs contained significant gaps and required substantial clarification.

math.DS↗
arXiv2025

Linear Flows on Translation Prisms

Motivated by the study of billiards in polyhedra, we study linear flows in a family of singular flat $3$-manifolds which we call translation prisms. Using ideas of Furstenberg and Veech, we connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. Furthermore, we construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in a right prism over a regular $n$-gons for $n=7, 9, 14, 16, 18, 20, 24, 30$.

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arXiv2024

The measure transfer for subshifts induced by a morphism of free monoids

Every non-erasing monoid morphism $σ: \mathcal{A}^* \to \mathcal{B}^*$ induces a {\em measure transfer map} $σ_X^{\mathcal{M}}: \mathcal{M}(X) \to \mathcal{M}(σ(X))$ between the measure cones $\mathcal{M}(X)$ and $\mathcal{M}(σ(X))$, associated to any subshift $X \subset \mathcal{A}^{\mathbb{Z}}$ and its image subshift $σ(X) \subset \mathcal{B}^{\mathbb{Z}}$ respectively. We define and study this map in detail and show that it is continuous, linear and functorial. It also turns out to be surjective \cite{BHL2.8-II}. Furthermore, an efficient technique to compute the value of the transferred measure $σ_X^{\mathcal{M}(μ)}$ on any cylinder $[w]$ (for $w \in \mathcal{B}^*$) is presented. \smallskip \noindent {\bf Theorem:} If a non-erasing morphism $σ: \mathcal{A}^* \to \mathcal{B}^*$ is injective on the shift-orbits of some subshift $X \subset \mathcal{A}^\mathbb{Z}$, then $σ^{\mathcal{M}_X}$ is injective. \smallskip The assumption on $σ$ that it is ``injective on the shift-orbits of $X$'' is strictly weaker than ``recognizable in $X$'', and strictly stronger than ``recognizable for aperiodic points in $X$''. The last assumption does in general not suffice to obtain the injectivity of the measure transfer map $σ_X^{\mathcal{M}}$.

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arXiv2023

Measure transfer and $S$-adic developments for subshifts

Based on previous work of the authors, to any $S$-adic development of a subshift $X$ a "directive sequence" of commutative diagrams is associated, which consists at every level $n \geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given $S$-adic development. The issuing rich picture enables one to deduce results about $X$ with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result we also exhibit, for any integer $d \geq 2$, an $S$-adic development of a minimal, aperiodic, uniquely ergodic subshift $X$, where all level alphabets ${\cal A}_n$ have cardinality $d\,$, while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n \subset {\cal A}_n^\mathbb Z$.

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arXiv2022

Piecewise rotations: limit set for the non -bijective maps

We consider non-bijective piecewise rotations of the plane. These maps belong to a family introduced in previous papers by Boshernitzan and Goetz. We derive in this paper some upper bounds to the size of the limit set. This improves results of \cite{Bosh.Goet.03}.

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arXiv2021

Density of Binary Disc Packings: The 9 Compact Packings

A disc packing in the plane is compact if its contact graph is a triangulation. There are $9$ values of $r$ such that a compact packing by discs of radii $1$ and $r$ exists. We prove, for each of these $9$ values, that the maximal density over all the packings by discs of radii $1$ and $r$ is reached for a compact packing (we give it as well as its density).

cs.DM↗
arXiv2020

Invariant measures on finite rank subshifts

In this note we show that for any subshift $X$ of finite $S$-rank every invariant measure $μ$ is determined by its values on finitely many cylinders. Under mild conditions these cylinders are given by the letters of the alphabet in question.

math.DS↗
arXiv2020

Symbolic dynamics for the piecewise rotations: Case of the bijective symmetric maps

We consider a specific %piecewise rotation of the plane that is continuous on two half-planes, class of piecewise rotations of the plane that are continuous on two half-planes, as studied in \cite{Bosh.Goet.03}, \cite{Goet.Quas.09} and \cite{Che.Goe.Qua.12}. %If Assuming that the angle belongs to the set $\{\fracπ{2},\fracπ{3},\fracπ{6},\fracπ{4}\} %$ $, we give a description of the symbolic dynamics of this map in the bijective symmetric case.

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arXiv2020

Tower power for $S$-adics

We explain and restate the results from our recent paper arXiv:1503.08000.v3 in standard language for substitutions and $S$-adic systems in symbolic dynamics. We then produce as rather direct application an $S$-adic system (with finite set of substitutions $S$ on $d$ letters) that is minimal and has $d$ distinct ergodic probability measures. As second application we exhibit a formula that allows an efficient practical computation of the cylinder measure $μ([w])$, for any word $w \in \cal A^*$ and any invariant measure $μ$ on the subshift $X_σ$ defined by any everywhere growing but not necessarily primitive or irreducible substitution $σ: \cal A^* \to \cal A^*$. Several examples are considered in detail, and model computations are presented.

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arXiv2019

Graph towers, laminations and their invariant measures

In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow Γ$ and vector towers $\overleftarrow v$ on $\overleftarrow Γ$, which allows us to efficiently describe all invariant measures $μ= μ^{\overleftarrow v}$ on any given shift space over a finite alphabet. The new technology admits a number of direct applications, in particular concerning invariant measures on non-primitive substitution subshifts, minimal subshifts with many ergodic measures, or an efficient calculation of the measure of a given cylinder. It also applies to currents on a free group $F_N$, and in particular the set of projectively fixed currents under the action of a (possibly reducible) endomorphism $φ: F_N \to F_N$ is determined, when $φ$ is represented by a train track map.

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arXiv2018

Canonical projection tilings defined by patterns

We give a necessary and sufficient condition on a $d$-dimensional affine subspace of $\mathbb{R}^n$ to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of their digitizations. The condition relies on the notion of {\em coincidence} and can be effectively checked. As a corollary, we get that only algebraic subspaces can be characterized by patterns.

math.DS↗
arXiv2017

Symbolic dynamics of a piecewise rotation: case of the non symmetric bijective maps

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 $\{\fracπ{2},\frac{2π}{3},\fracπ{4}\}$, we give a complete description of the symbolic dynamics of this map in the non symmetric bijective case.

math.DS↗
arXiv2017

Topological substitutions and Rauzy fractals

We consider two families of planar self-similar tilings of different nature: the tilings consisting of translated copies of the fractal sets defined by an iterated function system, and the tilings obtained as a geometrical realization of a topological substitution (an object of purely combinatorial nature). We establish a link between the two families in a specific case, by defining an explicit topological substitution and by proving that it generates the same tilings as those associated with the Tribonacci Rauzy fractal.

math.DS↗
arXiv2016

Weak local rules for planar octagonal tilings

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 90s.

math.DS↗
arXiv2016

Icosahedral Tilings Revisited

Icosahedral tilings, although non-periodic, are known to be characterized by their configurations of some finite size. This characterization has also been expressed in terms of a simple alternation condition. We provide an alternative proof - shorter and arguably easier - of this fact. We moreover conjecture that the alternation condition can be weakened.

math.CO↗