Benoît Loridant

Tilings induced by a class of cubic Rauzy fractals

We study aperiodic and periodic tilings induced by the Rauzy fractal and its subtiles associated with beta-substitutions related to the polynomial x^3-ax^2-bx-1 for a>=b>=1. In particular, we compute the corresponding boundary graphs, describing the adjacencies in the tilings. These graphs are a valuable tool for more advanced studies of the topological properties of the Rauzy fractals. As an example, we show that the Rauzy fractals are not homeomorphic to a closed disc as soon as a@?2b-4. The methods presented in this paper may be used to obtain similar results for other classes of substitutions.

A numerical scale for non-locally connected planar continua☆

Abstract We introduce a numerical scale to quantify to which extent a planar continuum is not locally connected. For a locally connected continuum, the numerical scale is zero; for a continuum like the topologists sine curve, the scale is one; for an indecomposable continuum, it is infinite. We use a purely topological framework of fibers and further characterize the local connectedness of a planar continuum in terms of triviality of its fibers.

INTERIOR COMPONENTS OF A TILE ASSOCIATED TO A QUADRATIC CANONICAL NUMBER SYSTEM — PART II

Let be a root of the polynomial p(x) = x2 + 4x + 5. It is well-known that the pair (α, {0, 1, 2, 3, 4}) forms a canonical number system, i.e., that each γ ∈ ℤ[α] admits a finite representation of the shape γ = a0 + a1α + ⋯ + alαl with ai ∈ {0, 1, 2, 3, 4}. The set of points with integer part 0 in this number system is called the fundamental domain of this canonical number system. It is a plane continuum with nonempty interior which induces a tiling of ℂ. Moreover, it has a disconnected interior . In the first paper of this series we described the closures C0, C1, C2 and C3 of the four largest components of as attractors of graph-directed self-similar sets. Each of these four sets is a translation of C0. We conjectured that the closures of the other components are images of C0 by similarity transformations. In this article we prove this conjecture. Moreover, we provide a graph from which the suitable similarities can be read off.

On a Theorem of Bandt and Wang and Its Extension to p2-tiles

We study tilings of the plane by a single prototile with respect to the lattice and to the crystallographic group p2. We are interested in the connection between the neighbors of a tile in the tiling and its topology. We show that lattice and p2-tiles always have at least six neighbors. We characterize self-affine tiles that are homeomorphic to a disk in a rather easy way by the set and number of neighbors of the central tile in the tiling. This extends the work of Bandt and Wang devoted to lattice self-affine disk-like tiles of the plane.

Rauzy fractals with countable fundamental group

We prove that every free group of finite rank can be realized as the fundamental group of a planar Rauzy fractal associated with a 4-letter unimodular cubic Pisot substitution. This characterizes all countable fundamental groups for planar Rauzy fractals. We give an explicit construction relying on two operations on substitutions: symbolic splittings and conjugations by free group automorphisms.

Geometrical Models for a Class of Reducible Pisot Substitutions

We set up a geometrical theory for the study of the dynamics of reducible Pisot substitutions. It is based on certain Rauzy fractals generated by duals of higher dimensional extensions of substitutions. We obtain under certain hypotheses geometric representations of stepped surfaces and related polygonal tilings, as well as self-replicating and periodic tilings made of Rauzy fractals. We apply our theory to one-parameter family of substitutions. For this family, we analyze and interpret in a new combinatorial way the codings of a domain exchange defined on the associated fractal domains. We deduce that the symbolic dynamical systems associated with this family of substitutions behave dynamically as first returns of toral translations.

Fundamental group of tiles associated to quadratic canonical number systems

If A is a 2 × 2 expanding matrix with integral coefficients, and ⊂ ℤ2 a complete set of coset representatives of ℤ2/Aℤ2 with |det(A)| elements, then the set ℐ defined by Aℐ = ℐ + is a self-affine plane tile of ℝ2, provided that its two-dimensional Lebesgue measure is positive.It was shown by Luo and Thuswaldner that the fundamental group of such a tile is either trivial or uncountable.To a quadratic polynomial x2 + Ax + B, A, B ∈ ℤ such that B ≥ 2 and −1 ≤ A ≤ B, one can attach a tile ℐ. Akiyama and Thuswaldner proved the triviality of the fundamental group of this tile for 2A < B + 3, by showing that a tile of this class is homeomorphic to a closed disk. The case 2A ≥ B + 3 is treated here by using the criterion given by Luo and Thuswaldner.