Topological properties of Rauzy fractals

Abstract

Abstract. -Substitutions are combinatorial objects (one replaces a letter by a word) which produce sequences by iteration. They occur in many mathematical fields, roughly as soon as a repetitive process appears. In the present monograph we deal with topological and geometric properties of substitutions, i.e., we study properties of the Rauzy fractals associated to substitutions.To be more precise, let σ be a substitution over the finite alphabet A. We assume that the incidence matrix of σ is primitive and its dominant eigenvalue is a unit Pisot number (i.e., an algebraic integer greater than one whose norm is equal to one and all of whose Galois conjugates are of modulus strictly smaller than one). It is well-known that one can attach to σ a set T which is now called central tile or Rauzy fractal of σ. Such a central tile is a compact set that is the closure of its interior and decomposes in a natural way in n = #A subtiles T (1), . . . , T (n). The central tile as well as its subtiles are graph directed self-affine sets that often have fractal boundary.Pisot substitutions and central tiles are of high relevance in several branches of mathematics like tiling theory, spectral theory, Diophantine approximation, the construction of discrete planes and quasicrystals as well as in connection with numeration like generalized continued fractions and radix representations. The questions coming up in all these domains can often be reformulated in terms of questions related to the topology and geometry of the underlying central tile.After a thorough survey of important properties of unit Pisot substitutions and their associated Rauzy fractals the present monograph is devoted to the investigation of a variety of topological properties of T and its subtiles. Our approach is an algorithmic one. In particular, we dwell upon the question whether T and its subtiles induce a tiling, calculate the Hausdorff dimension of their boundary, give criteria for their connectivity and homeomorphy to a disk and derive properties of their fundamental group.The basic tools for our criteria are several classes of graphs built from the description of the tiles T (i) (1 ≤ i ≤ n) as graph directed iterated function systems and from the structure of the tilings induced by these tiles. These graphs are of interest in their own right. For instance, they can be used to construct the boundaries ∂T as well as ∂T (i) (1 ≤ i ≤ n) and all points where two, three or four different tiles of the induced tilings meet.When working with central tiles in one of the above mentioned contexts it is often useful to know such intersection properties of tiles. In this sense the present monograph also aims at providing tools for "everyday's life" when dealing with topological and geometric properties of substitutions.Many examples are given throughout the text in order to illustrate our results. Moreover, we give perspectives for further directions of research related to the topics discussed in this monograph.

Linear Recursive Odometers and Beta-Expansions

“…This condition was introduced by Siegel and Thuswaldner [15] in the framework of tilings associated to Pisot substitutions, and for β-expansions in the present form in [12]. Moreover, if we allow sequences a 1 a 2 · · · that are not purely periodic, then Hypothesis B does not imply (F) but only (PF).…”

“…(1) is a left eigenvector of σ β , and L σ β = V β − V β Z , thus (QM) holds if and only if the β-substitution satisfies the quotient mapping condition of [15].…”

“…We first explain the relation of the condition (QM) above to the quotient mapping condition defined in [15] for substitutions. Let σ be a primitive substitution on a finite alphabet A and M σ = |σ(j)| i i,j∈A its incidence matrix, where |σ(j)| i denotes the number of occurrences of the letter i in σ(j).…”

“…A class of non-trivial examples of bases satisfying (QM) was given in [15,12] by β 3 = tβ 2 − β + 1, t ≥ 2; in this case, we have #V β = deg(β) + 1. The following theorem gives a characterization of (QM) for β > 1 satisfying…”

Balanced pair algorithm for a class of cubic substitutions

“…In particular, it plays a fundamental role in the Pisot conjecture. Geometrical and topological properties of Rauzy fractals have been studied extensively; see, among other references, [2,7,3,9,15,16,19,22]. It is a compact set equal to the closure of its interior and it decomposes naturally into 3 subtiles.…”