120 research outputs found
Random product of substitutions with the same incidence matrix
Any infinite sequence of substitutions with the same matrix of the Pisot type
defines a symbolic dynamical system which is minimal. We prove that, to any
such sequence, we can associate a compact set (Rauzy fractal) by projection of
the stepped line associated with an element of the symbolic system on the
contracting space of the matrix. We show that this Rauzy fractal depends
continuously on the sequence of substitutions, and investigate some of its
properties; in some cases, this construction gives a geometric model for the
symbolic dynamical system
Symmetric intersections of Rauzy fractals
In this article we study symmetric subsets of Rauzy fractals of unimodular
irreducible Pisot substitutions. The symmetry considered is reflection through
the origin. Given an unimodular irreducible Pisot substitution, we consider the
intersection of its Rauzy fractal with the Rauzy fractal of the reverse
substitution. This set is symmetric and it is obtained by the balanced pair
algorithm associated with both substitutions
The geometry of non-unit Pisot substitutions
Let be a non-unit Pisot substitution and let be the
associated Pisot number. It is known that one can associate certain fractal
tiles, so-called \emph{Rauzy fractals}, with . In our setting, these
fractals are subsets of a certain open subring of the ad\`ele ring
. We present several approaches on how to
define Rauzy fractals and discuss the relations between them. In particular, we
consider Rauzy fractals as the natural geometric objects of certain numeration
systems, define them in terms of the one-dimensional realization of
and its dual (in the spirit of Arnoux and Ito), and view them as the dual of
multi-component model sets for particular cut and project schemes. We also
define stepped surfaces suited for non-unit Pisot substitutions. We provide
basic topological and geometric properties of Rauzy fractals associated with
non-unit Pisot substitutions, prove some tiling results for them, and provide
relations to subshifts defined in terms of the periodic points of , to
adic transformations, and a domain exchange. We illustrate our results by
examples on two and three letter substitutions.Comment: 29 page
Bifix codes and interval exchanges
We investigate the relation between bifix codes and interval exchange
transformations. We prove that the class of natural codings of regular interval
echange transformations is closed under maximal bifix decoding.Comment: arXiv admin note: substantial text overlap with arXiv:1305.0127,
arXiv:1308.539
A class of cubic Rauzy Fractals
In this paper, we study arithmetical and topological properties for a class
of Rauzy fractals given by the polynomial
where is an integer. In particular, we prove the number of neighbors
of in the periodic tiling is equal to . We also give
explicitly an automaton that generates the boundary of . As a
consequence, we prove that is homeomorphic to a topological
disk
Fractal tiles associated with shift radix systems
Shift radix systems form a collection of dynamical systems depending on a
parameter which varies in the -dimensional real vector space.
They generalize well-known numeration systems such as beta-expansions,
expansions with respect to rational bases, and canonical number systems.
Beta-numeration and canonical number systems are known to be intimately related
to fractal shapes, such as the classical Rauzy fractal and the twin dragon.
These fractals turned out to be important for studying properties of expansions
in several settings. In the present paper we associate a collection of fractal
tiles with shift radix systems. We show that for certain classes of parameters
these tiles coincide with affine copies of the well-known tiles
associated with beta-expansions and canonical number systems. On the other
hand, these tiles provide natural families of tiles for beta-expansions with
(non-unit) Pisot numbers as well as canonical number systems with (non-monic)
expanding polynomials. We also prove basic properties for tiles associated with
shift radix systems. Indeed, we prove that under some algebraic conditions on
the parameter of the shift radix system, these tiles provide
multiple tilings and even tilings of the -dimensional real vector space.
These tilings turn out to have a more complicated structure than the tilings
arising from the known number systems mentioned above. Such a tiling may
consist of tiles having infinitely many different shapes. Moreover, the tiles
need not be self-affine (or graph directed self-affine)
Connectedness of fractals associated with Arnoux-Rauzy substitutions
Rauzy fractals are compact sets with fractal boundary that can be associated
with any unimodular Pisot irreducible substitution. These fractals can be
defined as the Hausdorff limit of a sequence of compact sets, where each set is
a renormalized projection of a finite union of faces of unit cubes. We exploit
this combinatorial definition to prove the connectedness of the Rauzy fractal
associated with any finite product of three-letter Arnoux-Rauzy substitutions.Comment: 15 pages, v2 includes minor corrections to match the published
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