70 research outputs found
Balancedness of Arnoux-Rauzy and Brun words
We study balancedness properties of words given by the Arnoux-Rauzy and Brun
multi-dimensional continued fraction algorithms. We show that almost all Brun
words on 3 letters and Arnoux-Rauzy words over arbitrary alphabets are finitely
balanced; in particular, boundedness of the strong partial quotients implies
balancedness. On the other hand, we provide examples of unbalanced Brun words
on 3 letters
Decidability Problems for Self-induced Systems Generated by a Substitution
International audienceIn this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties yields by the combinatorial substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems
-adic expansions related to continued fractions (Natural extension of arithmetic algorithms and S-adic system)
"Natural extension of arithmetic algorithms and S-adic system". July 20~24, 2015. edited by Shigeki Akiyama. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We consider S-adic expansions associated with continued fraction algorithms, where an S-adic expansion corresponds to an infinite composition of substitutions. Recall that a substitution is a morphism of the free monoid. We focus in particular on the substitutions associated with regular continued fractions (Sturmian substitutions), and with Arnoux-Rauzy, Brun, and Jacobi{Perron (multidimensional) continued fraction algorithms. We also discuss the spectral properties of the associated symbolic dynamical systems under a Pisot type assumption
Episturmian words: a survey
In this paper, we survey the rich theory of infinite episturmian words which
generalize to any finite alphabet, in a rather resembling way, the well-known
family of Sturmian words on two letters. After recalling definitions and basic
properties, we consider episturmian morphisms that allow for a deeper study of
these words. Some properties of factors are described, including factor
complexity, palindromes, fractional powers, frequencies, and return words. We
also consider lexicographical properties of episturmian words, as well as their
connection to the balance property, and related notions such as finite
episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize
the skew words of Morse and Hedlund.Comment: 36 pages; major revision: improvements + new material + more
reference
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
versio
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