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Julien Cassaigne

Publications and source records attributed to Julien Cassaigne.

At least 19 recordsLinked to original sources
arXiv2026

On modulo-recurrence and window complexity in infinite words

In this paper, we introduce the notions of uniform modulo-recurrence and strong modulo-recurrence. We show that Sturmian words and maximal complexity words are strongly modulo-recurrent. Next, a relationship between the window complexity and the classical complexity of the Thue-Morse word is established. We provide an aperiodic recurrent word such that the window complexity is bounded. We also construct a family of aperiodic recurrent words such that their window complexity $P^w(n)$ is in $O(n^\alpha)$, while at least $n^\alpha$ for infinitely many $n$, where $0<\alpha<1$. Finally, we establish that the window complexity of a uniformly recurrent aperiodic word is unbounded.

math.CO↗
arXiv2026

The complexity of smooth words over binary alphabets

Smooth words over an alphabet of non-negative integers $\{a,b\}$ are infinite words that are infinitely derivable, the emblematic example being the Oldenburger-Kolakoski word over $\{1,2\}$. The main way to study their language is to consider a finite version of smooth words that we call f-smooth words. In this paper we prove that the f-smooth words are exactly the factors of smooth words, and we make progress towards the conjecture of Sing that the complexity of f-smooth words over $\{a,b\}$ grows like $\Theta\left(n^{\log(a+b)/\log((a+b)/2)}\right)$: we prove it over even alphabets, we prove the lower bound over any binary alphabet and we improve the known upper bound over odd alphabets.

cs.FL↗
arXiv2026

Symbols frequencies in the Thue--Morse word in base $3/2$ and related conjectures

We study a binary Thue--Morse-type sequence arising from the base-$3/2$ expansion of integers, an archetypal automatic sequence in a rational base numeration system. Because the sequence is generated by a periodic iteration of morphisms rather than a single primitive substitution, classical Perron--Frobenius methods do not directly apply to determine symbol frequencies. We prove that both symbols ${\tt 0},{\tt 1}$ occur with frequency $1/2$ and we show uniform recurrence and symmetry properties of its set of factors. The proof reveals a structural bridge between combinatorics on words and harmonic analysis: the first difference sequence is shown to be Toeplitz, providing dynamical rigidity, while filtered frequencies naturally encode a dyadic structure that lifts to the compact group of $2$-adic integers. In this $2$-adic setting, desubstitution becomes a linear operator on Fourier coefficients, and a spectral contraction argument enforces uniqueness of limiting densities. Our results answer several conjectures of Dekking (on a sibling sequence) and illustrate how harmonic analysis on compact groups can be fruitfully combined with substitution dynamics.

math.CO↗
arXiv2025

Freezing phase transition for the Thue-Morse subshift

On the full shift on two symbols, we consider the potential defined by $V(x) = \frac{1}{n}$ where $n$ denotes the longest common prefix between the infinite word $x$ and an element of the subshift associated to the Thue-Morse substitution. Given a non negative real number $\beta$, the pressure function is $P(\beta):=\sup\left\{h_{\mu}+\beta\int V\,d\mu\right\},$ where the supremum is taken over all shift invariant probabilities $\mu$ on the full shift and $h_{\mu}$ is the Kolmogorov entropy. We prove that there is a freezing phase transition for the potential $V$: For $\beta$ large enough, the pressure $P(\be)$ is equal to zero. Similar results were previously published by Bruin and Leplaideur in \cite{BL2}, \cite{Bruin-Leplaid-13} but their proofs contained significant gaps and required substantial clarification.

math.DS↗
arXiv2022

Word of low complexity without uniform frequencies

In this paper, we construct a uniformely recurrent infinite word of low complexity without uniform frequencies of letters. This shows the optimality of a bound of Boshernitzan, which gives a sufficient condition for a uniformly recurrent infinite word to admit uniform frequencies.

math.DS↗
arXiv2022

Almost everywhere balanced sequences of complexity $2n+1$

We study ternary sequences associated with a multidimensional continued fraction algorithm introduced by the first author. The algorithm is defined by two matrices and we show that it is measurably isomorphic to the shift on the set $\{1,2\}^\mathbb{N}$ of directive sequences. For a given set $\mathcal{C}$ of two substitutions, we show that there exists a $\mathcal{C}$-adic sequence for every vector of letter frequencies or, equivalently, for every directive sequence. We show that their factor complexity is at most $2n+1$ and is $2n+1$ if and only if the letter frequencies are rationally independent if and only if the $\mathcal{C}$-adic representation is primitive. It turns out that in this case, the sequences are dendric. We also prove that $μ$-almost every $\mathcal{C}$-adic sequence is balanced, where $μ$ is any shift-invariant ergodic Borel probability measure on $\{1,2\}^\mathbb{N}$ giving a positive measure to the cylinder $[12121212]$. We also prove that the second Lyapunov exponent of the matrix cocycle associated with the measure $μ$ is negative.

math.DS↗
arXiv2019

Arnoux-Rauzy interval exchange transformations

The Arnoux-Rauzy systems are defined in \cite{ar}, both as symbolic systems on three letters and exchanges of six intervals on the circle. In connection with a conjecture of S.P. Novikov, we investigate the dynamical properties of the interval exchanges, and precise their relation with the symbolic systems, which was known only to be a semi-conjugacy; in order to do this, we define a new system which is an exchange of nine intervals on the line (it was described in \cite{abb} for a particular case). Our main result is that the semi-conjugacy determines a measure-theoretic isomorphism (between the three systems) under a diophantine (sufficient) condition, which is satisfied by almost all Arnoux-Rauzy systems for a suitable measure; but, under another condition, the interval exchanges are not uniquely ergodic and the isomorphism does not hold for all invariant measures; finally, we give conditions for these interval exchanges to be weakly mixing.

math.DS↗
arXiv2017

A Taxonomy of Morphic Sequences

In this note we classify sequences according to whether they are morphic, pure morphic, uniform morphic, pure uniform morphic, primitive morphic, or pure primitive morphic, and for each possibility we either give an example or prove that no example is possible.

cs.FL↗
arXiv2017

On the computational complexity of algebraic numbers: the Hartmanis--Stearns problem revisited

We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. We show that the Hartmanis--Stearns problem can be solved in a satisfactory way for the class of multistack machines. In this direction, our main result is that the base-$b$ expansion of an algebraic irrational real number cannot be generated by a deterministic pushdown automaton. We also confirm an old claim of Cobham proving that such numbers cannot be generated by a tag machine with dilation factor larger than one.

math.NT↗
arXiv2017

A Set of Sequences of Complexity $2n+1$

We prove the existence of a ternary sequence of factor complexity $2n+1$ for any given vector of rationally independent letter frequencies. Such sequences are constructed from an infinite product of two substitutions according to a particular Multidimensional Continued Fraction algorithm. We show that this algorithm is conjugate to a well-known one, the Selmer algorithm. Experimentations (Baldwin, 1992) suggest that their second Lyapunov exponent is negative which presages finite balance properties.

math.CO↗
arXiv2016

Cost and dimension of words of zero topological entropy

Let $A^*$ denote the free monoid generated by a finite nonempty set $A.$ In this paper we introduce a new measure of complexity of languages $L\subseteq A^*$ defined in terms of the semigroup structure on $A^*.$ For each $L\subseteq A^*,$ we define its {\it cost} $c(L)$ as the infimum of all real numbers $α$ for which there exist a language $S\subseteq A^*$ with $p_S(n)=O(n^α)$ and a positive integer $k$ with $L\subseteq S^k.$ We also define the {\it cost dimension} $d_c(L)$ as the infimum of the set of all positive integers $k$ such that $L\subseteq S^k$ for some language $S$ with $p_S(n)=O(n^{c(L)}).$ We are primarily interested in languages $L$ given by the set of factors of an infinite word $x=x_0x_1x_2\cdots \in A^ω$ of zero topological entropy, in which case $c(L)<+\infty.$ We establish the following characterisation of words of linear factor complexity: Let $x\in A^ω$ and $L=$Fac$(x)$ be the set of factors of $x.$ Then $p_x(n)=Θ(n)$ if and only $c(L)=0$ and $d_c(L)=2.$ In other words, $p_x(n)=O(n)$ if and only if Fac$(x)\subseteq S^2$ for some language $S\subseteq A^+$ of bounded complexity (meaning $\limsup p_S(n)<+\infty).$ In general the cost of a language $L$ reflects deeply the underlying combinatorial structure induced by the semigroup structure on $A^*.$ For example, in contrast to the above characterisation of languages generated by words of sub-linear complexity, there exist non factorial languages $L$ of complexity $p_L(n)=O(\log n)$ (and hence of cost equal to $0)$ and of cost dimension $+\infty.$ In this paper we investigate the cost and cost dimension of languages defined by infinite words of zero topological entropy.

math.CO↗
arXiv2016

Cyclic Complexity of Words

We introduce and study a complexity function on words $c_x(n),$ called \emph{cyclic complexity}, which counts the number of conjugacy classes of factors of length $n$ of an infinite word $x.$ We extend the well-known Morse-Hedlund theorem to the setting of cyclic complexity by showing that a word is ultimately periodic if and only if it has bounded cyclic complexity. Unlike most complexity functions, cyclic complexity distinguishes between Sturmian words of different slopes. We prove that if $x$ is a Sturmian word and $y$ is a word having the same cyclic complexity of $x,$ then up to renaming letters, $x$ and $y$ have the same set of factors. In particular, $y$ is also Sturmian of slope equal to that of $x.$ Since $c_x(n)=1$ for some $n\geq 1$ implies $x$ is periodic, it is natural to consider the quantity $\liminf_{n\rightarrow \infty} c_x(n).$ We show that if $x$ is a Sturmian word, then $\liminf_{n\rightarrow \infty} c_x(n)=2.$ We prove however that this is not a characterization of Sturmian words by exhibiting a restricted class of Toeplitz words, including the period-doubling word, which also verify this same condition on the limit infimum. In contrast we show that, for the Thue-Morse word $t$, $\liminf_{n\rightarrow \infty} c_t(n)=+\infty.$

cs.FL↗
arXiv2014

Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More

We study the decidability of three well-known problems related to integer matrix multiplication: Mortality (M), Zero in the Left-Upper Corner (Z), and Zero in the Right-Upper Corner (R). Let d and k be positive integers. Define M(k, d x d) as the following special case of the Mortality problem: given a set X of d -by-d integer matrices such that the cardinality of X is not greater than k, decide whether the d-by-d zero matrix belongs to X^+, where X^+ denotes the closure of X under the usual matrix multiplication. In the same way, define the Z(k, d x d) problem as: given an instance X of M(k, d x d) (the instances of Z(k, d x d) are the same as those of M(k, d x d)), decide whether at least one matrix in X^+ has a zero in the left-upper corner. Define R(k, d x d) as the variant of Z(k, d x d) where "left-upper corner" is replaced with "right-upper corner". In the paper, we prove that M(6, 3 x 3), M(4, 5 x 5), M(3, 9 x 9), M(2, 15 x 15), Z(5, 3 x 3), Z(3, 5 x 5), Z(2, 9 x 9), R(6, 3 x 3), R(5, 4 x 4), and R(3, 6 x 6) are undecidable. The previous best comparable results were the undecidabilities of M(7, 3 x 3), M(3, 13 x 13), M(2, 21 x 21), Z(7, 3 x 3), Z(2, 13 x 13), R(7, 3 x 3), and R(2, 10 x 10).

cs.DM↗
arXiv2013

Invariant games and non-homogeneous Beatty sequences

We characterize all the pairs of complementary non-homogenous Beatty sequences $(A_n)_{n\ge 0}$ and $(B_n)_{n\ge 0}$ for which there exists an invariant game having exactly $\{(A_n,B_n)\mid n\ge 0\}\cup \{(B_n,A_n)\mid n\ge 0\}$ as set of $\mathcal{P}$-positions. Using the notion of Sturmian word and tools arising in symbolic dynamics and combinatorics on words, this characterization can be translated to a decision procedure relying only on a few algebraic tests about algebraicity or rational independence. Given any four real numbers defining the two sequences, up to these tests, we can therefore decide whether or not such an invariant game exists.

math.CO↗
arXiv2012

Balance properties of Arnoux-Rauzy words

The paper deals with balances and imbalances in Arnoux-Rauzy words. We provide sufficient conditions for $C$-balancedness, but our results indicate that even a characterization of 2-balanced Arnoux-Rauzy words on a 3-letter alphabet is not immediate.

cs.FL↗
arXiv2012

On the decidability of semigroup freeness

This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X of S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoid. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over three-by-three integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is three-fold: (i) to present general results concerning freeness problems, (ii) to study the decidability of freeness problems over various particular semigroups (special attention is devoted to multiplicative matrix semigroups), and (iii) to propose precise, challenging open questions in order to promote the study of the topic.

cs.DM↗
arXiv2012

On the Morse-Hedlund complexity gap

In 1938, Morse and Hedlund proved that the subword complexity function of an infinite word is either bounded or at least linearly growing. In 1982, Ehrenfeucht and Rozenberg proved that this gap property holds for the subword complexity function of any language. The aim of the present paper is to present a self-contained, compact proof of Ehrenfeucht and Rozenberg's result.

cs.FL↗