### About some Riemann surfaces and plane tilings.

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The atomic surfaces of unimodular Pisot substitutions of irreducible type have been studied by many authors. In this article, we study the atomic surfaces of Pisot substitutions of reducible type.As an analogue of the irreducible case, we define the stepped-surface and the dual substitution over it. Using these notions, we give a simple proof to the fact that atomic surfaces form a self-similar tiling system. We show that the stepped-surface possesses the quasi-periodic property, which implies that...

The Fourier transform of a weighted Dirac comb of beta-integers is characterized within the framework of the theory of Distributions, in particular its pure point part which corresponds to the Bragg part of the diffraction spectrum. The corresponding intensity function on this Bragg part is computed. We deduce the diffraction spectrum of weighted Delone sets on beta-lattices in the split case for the weight, when beta is the golden mean.

We show that an aperiodic minimal tiling space with only finitely many asymptotic composants embeds in a surface if and only if it is the suspension of a symbolic interval exchange transformation (possibly with reversals). We give two necessary conditions for an aperiodic primitive substitution tiling space to embed in a surface. In the case of substitutions on two symbols our classification is nearly complete. The results characterize the codimension one hyperbolic attractors of surface diffeomorphisms...

We investigate in a geometrical way the point sets of $\mathbb{R}$ obtained by the $\beta $-numeration that are the $\beta $-integers ${\mathbb{Z}}_{\beta}\subset \mathbb{Z}\left[\beta \right]$ where $\beta $ is a Perron number. We show that there exist two canonical cut-and-project schemes associated with the $\beta $-numeration, allowing to lift up the $\beta $-integers to some points of the lattice ${\mathbb{Z}}^{m}$ ($m=$ degree of $\beta $) lying about the dominant eigenspace of the companion matrix of $\beta $ . When $\beta $ is in particular a Pisot number, this framework gives another proof of the fact that ${\mathbb{Z}}_{\beta}$ is...

In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of $2D$-gons (hexagons, octagons, decagons, etc.). We show that the sets of partitions, ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of a $2D$-gon is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical models exist.

In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of 2D-gons (hexagons, octagons, decagons, etc.). We show that the sets of partitions, ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of a 2D-gon is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical models exist.

Let $\beta \>1$ be an algebraic number. We study the strings of zeros (“gaps”) in the Rényi $\beta $-expansion ${d}_{\beta}\left(1\right)$ of unity which controls the set ${\mathbb{Z}}_{\beta}$ of $\beta $-integers. Using a version of Liouville’s inequality which extends Mahler’s and Güting’s approximation theorems, the strings of zeros in ${d}_{\beta}\left(1\right)$ are shown to exhibit a “gappiness” asymptotically bounded above by $log\left(\mathrm{M}\right(\beta \left)\right)/log\left(\beta \right)$, where $\mathrm{M}\left(\beta \right)$ is the Mahler measure of $\beta $. The proof of this result provides in a natural way a new classification of algebraic numbers $\>1$ with classes called Q...

The spectrum of a weighted Dirac comb on the Thue-Morse quasicrystal is investigated by means of the Bombieri-Taylor conjecture, for Bragg peaks, and of a new conjecture that we call Aubry-Godrèche-Luck conjecture, for the singular continuous component. The decomposition of the Fourier transform of the weighted Dirac comb is obtained in terms of tempered distributions. We show that the asymptotic arithmetics of the $p$-rarefied sums of the Thue-Morse sequence (Dumont; Goldstein, Kelly and Speer; Grabner;...

In this paper we study multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We show for a large class of substitutions that the resulting word is the restriction of a linear function modulo $1$ and that it can be decided whether the resulting word is space filling or not. The proof uses lattices and the abstract number system associated with the substitution.