38 research outputs found
Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers
We study arithmetical and combinatorial properties of -integers for
being the root of the equation . We determine with the accuracy of the maximal number of
-fractional positions, which may arise as a result of addition of two
-integers. For the infinite word coding distances between
consecutive -integers, we determine precisely also the balance. The word
is the fixed point of the morphism and . In the case the corresponding infinite word is
sturmian and therefore 1-balanced. On the simplest non-sturmian example with
, we illustrate how closely the balance and arithmetical properties of
-integers are related.Comment: 15 page
Integers in number systems with positive and negative quadratic Pisot base
We consider numeration systems with base and , for quadratic
Pisot numbers and focus on comparing the combinatorial structure of the
sets and of numbers with integer expansion in base
, resp. . Our main result is the comparison of languages of
infinite words and coding the ordering of distances
between consecutive - and -integers. It turns out that for a
class of roots of , the languages coincide, while for other
quadratic Pisot numbers the language of can be identified only with
the language of a morphic image of . We also study the group
structure of -integers.Comment: 19 pages, 5 figure
Repetitions in beta-integers
Classical crystals are solid materials containing arbitrarily long periodic
repetitions of a single motif. In this paper, we study the maximal possible
repetition of the same motif occurring in beta-integers -- one dimensional
models of quasicrystals. We are interested in beta-integers realizing only a
finite number of distinct distances between neighboring elements. In such a
case, the problem may be reformulated in terms of combinatorics on words as a
study of the index of infinite words coding beta-integers. We will solve a
particular case for beta being a quadratic non-simple Parry number.Comment: 11 page
Return Words and Recurrence Function of a Class of Infinite Words
Many combinatorial and arithmetical properties have been studied for infinite words ub associated with Ă-integers. Here, new results describing return words and recurrence function for a special case of ub will be presented. The methods used here can be applied to more general infinite words, but the description then becomes rather technical.
Abelian Complexity of Infinite Words Associated with Quadratic Parry Numbers
We derive an explicit formula for the Abelian complexity of infinite words
associated with quadratic Parry numbers.Comment: 12 page
Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words
A word defined over an alphabet is -balanced
() if for all pairs of factors , of of the same
length and for all letters , the difference between the number
of letters in and is less or equal to . In this paper we
consider a ternary alphabet and a class of
substitutions defined by , ,
where . We prove that the fixed point of ,
formally written as , is 3-balanced and that its Abelian
complexity is bounded above by the value 7, regardless of the value of . We
also show that both these bounds are optimal, i.e. they cannot be improved.Comment: 26 page
Nested quasicrystalline discretisations of the line
One-dimensional cut-and-project point sets obtained from the square lattice
in the plane are considered from a unifying point of view and in the
perspective of aperiodic wavelet constructions. We successively examine their
geometrical aspects, combinatorial properties from the point of view of the
theory of languages, and self-similarity with algebraic scaling factor
. We explain the relation of the cut-and-project sets to non-standard
numeration systems based on . We finally examine the substitutivity, a
weakened version of substitution invariance, which provides us with an
algorithm for symbolic generation of cut-and-project sequences
On the growth of cocompact hyperbolic Coxeter groups
For an arbitrary cocompact hyperbolic Coxeter group G with finite generator
set S and complete growth function P(x)/Q(x), we provide a recursion formula
for the coefficients of the denominator polynomial Q(x) which allows to
determine recursively the Taylor coefficients and the pole behavior of the
growth function of G in terms of its Coxeter subgroup structure. We illustrate
this in the easy case of compact right-angled hyperbolic n-polytopes. Finally,
we provide detailed insight into the case of Coxeter groups with at most 6
generators, acting cocompactly on hyperbolic 4-space, by considering the three
combinatorially different families discovered and classified by Lanner,
Kaplinskaya and Esselmann, respectively.Comment: 24 page
Dynamical Directions in Numeration
International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper