Fractal representation of the attractive lamination of an automorphism of the free group

Arnoux, Pierre; Berthé, Valérie; Hilion, Arnaud; Siegel, Anne

doi:10.5802/aif.2237

Cited by 17 publications

(24 citation statements)

References 37 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A specific case is given by homeomorphisms of orientable surfaces with nonempty boundary: the homeomorphism of the surface can be coded into an automorphism of the homotopy group of the surface, which is called geometrical. Notice that even if all automorphisms of the group free group of rank two F 2 are geometrical, most automorphisms of free groups are not: for instance, the Tribonacci automorphism 1 → 12, 2 → 13, 3 → 1 is not geometrical in the free group of rank three and more generally, no irreducible automorphism on a free group of odd rank comes from a homeomorphism of an orientable surface [27].…”

Section: Invariants In Dynamics and Geometrymentioning

confidence: 99%

“…A natural question then becomes to identify substitutions that produce (i.e, they are sections of) the same tiling flow. However, several pictures showed that the central tiles for substitutions that are conjugate to each other look globally the same (see [27]). Therefore, a related challenge is to check which topological properties are invariant under the action of invertible substitutions.…”

Section: Invariants In Dynamics and Geometrymentioning

confidence: 99%

“…Introducing the symbolic representation of an attractive lamination by means of a train track allows to geometrically represent the lamination by a central tile, as soon as the automorphism has a unit Pisot dilation coefficient [27]. However, the construction depends on the train-track used to represent the automorphism; additionally automorphisms are considered in this context up to conjugacy by inner automorphisms.…”

Section: Invariants In Dynamics and Geometrymentioning

confidence: 99%

“…They describe an algorithmic process to build a representative for the automorphism, called an improved relative train-track mapping, which takes care of cancellations so that one can build a reduced two-sided recurrent infinite word on which the automorphism acts without cancellation and which is fixed by some power of the automorphism. From this, one deduces a symbolic dynamical system that is proved to be a representation of the attractive algebraic lamination [27].…”

Section: Invariants In Dynamics and Geometrymentioning

confidence: 99%

See 3 more Smart Citations

Topological properties of Rauzy fractals

Siegel¹,

Thuswaldner²

2009

Mémoires Soc. Math. France

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Invariants In Dynamics and Geometrymentioning

confidence: 99%

Section: Invariants In Dynamics and Geometrymentioning

confidence: 99%

Section: Invariants In Dynamics and Geometrymentioning

confidence: 99%

Section: Invariants In Dynamics and Geometrymentioning

confidence: 99%

See 2 more Smart Citations

Topological properties of Rauzy fractals

Siegel¹,

Thuswaldner²

2009

Mémoires Soc. Math. France

Self Cite

View full text Add to dashboard Cite

show abstract

“…Rauzy fractals can more generally be associated with Pisot substitutions (see [55,93,94,190,253,254,306,307] and the surveys [65,272]), as well as with Pisot β-shifts under the name of central tiles (see [7,8,9,10]), but they also can be associated with abstract numeration systems [62], as well as with some automorphisms of the free group [30], namely the so-called irreducible with irreducible powers automorphisms [70].…”

Section: Rauzy Fractalsmentioning

confidence: 99%