Substitutions, Rauzy fractals and tilings.

*(English)*Zbl 1247.37015
Berthé, Valérie (ed.) et al., Combinatorics, automata, and number theory. Cambridge: Cambridge University Press (ISBN 978-0-521-51597-9/hbk). Encyclopedia of Mathematics and its Applications 135, 248-323 (2010).

Summary: This chapter focuses on multiple tilings associated with substitutive dynamical systems. It is is organised as follows.

Section 5.2 gathers all the introductory material. We assume that we are given a unit Pisot irreducible substitution \(\sigma\). A suitable decomposition of the space \(\mathbb R^{n-1}\) is first introduced in Section 5.2.1 with respect to the eigenspaces of the incidence matrix \(\mathbf M_\sigma\) of \(\sigma\). A definition of the central tile associated with \(\sigma\) as well as its decomposition into subtiles is then provided in Section 5.2.2. We discuss the graph-directed set equation satisfied by the subtiles in Section 5.2.3. Two (multiple) tilings associated with \(\sigma\) are then introduced in Section 5.3. The first one, introduced in Section 5.3.2, is called tiling of the expanding line. This tiling by intervals tiles the expanding line of the incidence matrix \(\mathbf M_\sigma\) of \(\sigma\). The second one is a priori not a tiling, but a multiple tiling. It is defined on the contracting space of the incidence matrix \(\mathbf M_\sigma\), and it is made of translated copies of the subtiles of the central tile. It is called the self-replicating multiple tiling. Note that it is conjectured to be a tiling. It will be the main objective of the present chapter to introduce various graphs that provide conditions for this multiple tiling to be a tiling.

The first series of tiling conditions is expressed in geometric terms directly related to properties of the self-replicating multiple tiling. We start in Section 5.4.1 with a sufficient tiling property inspired by the so-called finiteness property (F) (discussed in Section 2.3.2.2). This leads us to introduce successively several graphs in Section 5.4 and Section 5.5, yielding necessary and sufficient conditions. We then discuss in Section 5.6, 5.7 and 5.8 further formulations for the tiling property expressed in terms of the tiling of the expanding line. They can be considered as dual to the former set of conditions. In particular, a formulation in terms of the so-called overlap coincidence condition is provided in Section 5.7, as well as, in Section 5.8, a further effective condition based on the notion of balanced pairs.

For the entire collection see [Zbl 1197.68006].

Section 5.2 gathers all the introductory material. We assume that we are given a unit Pisot irreducible substitution \(\sigma\). A suitable decomposition of the space \(\mathbb R^{n-1}\) is first introduced in Section 5.2.1 with respect to the eigenspaces of the incidence matrix \(\mathbf M_\sigma\) of \(\sigma\). A definition of the central tile associated with \(\sigma\) as well as its decomposition into subtiles is then provided in Section 5.2.2. We discuss the graph-directed set equation satisfied by the subtiles in Section 5.2.3. Two (multiple) tilings associated with \(\sigma\) are then introduced in Section 5.3. The first one, introduced in Section 5.3.2, is called tiling of the expanding line. This tiling by intervals tiles the expanding line of the incidence matrix \(\mathbf M_\sigma\) of \(\sigma\). The second one is a priori not a tiling, but a multiple tiling. It is defined on the contracting space of the incidence matrix \(\mathbf M_\sigma\), and it is made of translated copies of the subtiles of the central tile. It is called the self-replicating multiple tiling. Note that it is conjectured to be a tiling. It will be the main objective of the present chapter to introduce various graphs that provide conditions for this multiple tiling to be a tiling.

The first series of tiling conditions is expressed in geometric terms directly related to properties of the self-replicating multiple tiling. We start in Section 5.4.1 with a sufficient tiling property inspired by the so-called finiteness property (F) (discussed in Section 2.3.2.2). This leads us to introduce successively several graphs in Section 5.4 and Section 5.5, yielding necessary and sufficient conditions. We then discuss in Section 5.6, 5.7 and 5.8 further formulations for the tiling property expressed in terms of the tiling of the expanding line. They can be considered as dual to the former set of conditions. In particular, a formulation in terms of the so-called overlap coincidence condition is provided in Section 5.7, as well as, in Section 5.8, a further effective condition based on the notion of balanced pairs.

For the entire collection see [Zbl 1197.68006].

##### MSC:

37B50 | Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) |

11B83 | Special sequences and polynomials |