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Julien Leroy

Publications and source records attributed to Julien Leroy.

24 records · Page 2Linked to original sources
arXiv2015

The finite index basis property

We describe in this paper a connection between bifix codes, symbolic dynamical systems and free groups. This is in the spirit of the connection established previously for the symbolic systems corresponding to Sturmian words. We introduce a class of sets of factors of an infinite word with linear factor complexity containing Sturmian sets and regular interval exchange sets, namemly the class of tree sets. We prove as a main result that for a uniformly recurrent tree set $F$, a finite bifix code $X$ on the alphabet $A$ is $F$-maximal of $F$-degree $d$ if and only if it is the basis of a subgroup of index $d$ of the free group on $A$.

cs.DM↗
arXiv2015

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.

math.CO↗
arXiv2013

An analogue of Cobham's theorem for graph directed iterated function systems

Feng and Wang showed that two homogeneous iterated function systems in $\mathbb{R}$ with multiplicatively independent contraction ratios necessarily have different attractors. In this paper, we extend this result to graph directed iterated function systems in $\mathbb{R}^n$ with contraction ratios that are of the form $\frac{1}β$, for integers $β$. By using a result of Boigelot et al., this allows us to give a proof of a conjecture of Adamczewski and Bell. In doing so, we link the graph directed iterated function systems to Büchi automata. In particular, this link extends to real numbers $β$. We introduce a logical formalism that permits to characterize sets of $\mathbb{R}^n$ whose representations in base $β$ are recognized by some Büchi automata. This result depends on the algebraic properties of the base: $β$ being a Pisot or a Parry number. The main motivation of this work is to draw a general picture representing the different frameworks where an analogue of Cobham's theorem is known.

math.DS↗
arXiv2013

An $S$-adic characterization of minimal subshifts with first difference of complexity $1 \leq p(n+1) - p(n) \leq 2$

In [Ergodic Theory Dynam. System, 16 (1996) 663--682], S. Ferenczi proved that any minimal subshift with first difference of complexity bounded by 2 is $S$-adic with $\card S \leq 3^{27}$. In this paper, we improve this result by giving an $S$-adic charaterization of these subshifts with a set $S$ of 5 morphisms, solving by this way the $S$-adic conjecture for this particular case.

cs.DM↗
arXiv2012

Towards a statement of the S-adic conjecture through examples

The $S$-adic conjecture claims that there exists a condition $C$ such that a sequence has a sub-linear complexity if and only if it is an $S$-adic sequence satisfying Condition $C$ for some finite set $S$ of morphisms. We present an overview of the factor complexity of $S$-adic sequences and we give some examples that either illustrate some interesting properties or that are counter-examples to what could be believed to be "a good Condition $C$".

cs.DM↗