253 research outputs found
Beta-expansions, natural extensions and multiple tilings associated with Pisot units
From the works of Rauzy and Thurston, we know how to construct (multiple)
tilings of some Euclidean space using the conjugates of a Pisot unit
and the greedy -transformation. In this paper, we consider different
transformations generating expansions in base , including cases where
the associated subshift is not sofic. Under certain mild conditions, we show
that they give multiple tilings. We also give a necessary and sufficient
condition for the tiling property, generalizing the weak finiteness property
(W) for greedy -expansions. Remarkably, the symmetric
-transformation does not satisfy this condition when is the
smallest Pisot number or the Tribonacci number. This means that the Pisot
conjecture on tilings cannot be extended to the symmetric
-transformation. Closely related to these (multiple) tilings are natural
extensions of the transformations, which have many nice properties: they are
invariant under the Lebesgue measure; under certain conditions, they provide
Markov partitions of the torus; they characterize the numbers with purely
periodic expansion, and they allow determining any digit in an expansion
without knowing the other digits
Linear recursive odometers and beta-expansions
The aim of this paper is to study the connection between different properties
related to -expansions. In particular, the relation between two
conditions, both ensuring pure discrete spectrum of the odometer, is analysed.
The first one is the so-called Hypothesis B for the -odometers and the
second one is denoted by (QM) and it has been introduced in the framework of
tilings associated to Pisot -numerations
Purely periodic beta-expansions in the Pisot non-unit case
It is well known that real numbers with a purely periodic decimal expansion
are the rationals having, when reduced, a denominator coprime with 10. The aim
of this paper is to extend this result to beta-expansions with a Pisot base
beta which is not necessarily a unit: we characterize real numbers having a
purely periodic expansion in such a base; this characterization is given in
terms of an explicit set, called generalized Rauzy fractal, which is shown to
be a graph-directed self-affine compact subset of non-zero measure which
belongs to the direct product of Euclidean and p-adic spaces
Shift Radix Systems - A Survey
Let be an integer and . The {\em shift radix system} is defined by has the {\em finiteness
property} if each is eventually mapped to
under iterations of . In the present survey we summarize
results on these nearly linear mappings. We discuss how these mappings are
related to well-known numeration systems, to rotations with round-offs, and to
a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review
the behavior of the orbits of points under iterations of with
special emphasis on ultimately periodic orbits and on the finiteness property.
We also describe a geometric theory related to shift radix systems.Comment: 45 pages, 16 figure
Arithmetic Dynamics
This survey paper is aimed to describe a relatively new branch of symbolic
dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic
expansions of reals and vectors that have a "dynamical" sense. This means
precisely that they (semi-) conjugate a given continuous (or
measure-preserving) dynamical system and a symbolic one. The classes of
dynamical systems and their codings considered in the paper involve: (1)
Beta-expansions, i.e., the radix expansions in non-integer bases; (2)
"Rotational" expansions which arise in the problem of encoding of irrational
rotations of the circle; (3) Toral expansions which naturally appear in
arithmetic symbolic codings of algebraic toral automorphisms (mostly
hyperbolic).
We study ergodic-theoretic and probabilistic properties of these expansions
and their applications. Besides, in some cases we create "redundant"
representations (those whose space of "digits" is a priori larger than
necessary) and study their combinatorics.Comment: 45 pages in Latex + 3 figures in ep
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