arXiv subjects

Léo Poirier

Publications and source records attributed to Léo Poirier.

3 recordsLinked to original sources

arXiv2026

Frequency of patterns in smooth sequences over the alphabet \{1, 3\}

We provide an ergodic theory framework to study statistical properties of smooth sequences over the odd alphabet {1, 3}. The arithmetic nature of this alphabet yields a partition of the subshift of smooth sequences based on their local structure, defining a notion of type for those sequences. We describe the substitutive structure of the smaller subshifts obtained by fixing the sequence of types of the successive derivatives of smooth sequences, from which we obtain the unique ergodicity of all these subshifts. A direct consequence is that the asymptotic frequency of any finite pattern in a smooth sequence over {1, 3} is always well-defined and depends on its type sequence. Finally, we characterize the minimality of these subshifts, and propose some perspectives.

math.DS↗

arXiv2026

Frequencies of Patterns in Smooth Sequences Over the Alphabet $\{1,3\}$

We provide an ergodic theory framework to study statistical properties of smooth sequences over the odd alphabet {1,3}. The arithmetic nature of this alphabet yields a partition of the subshift of smooth sequences based on their local structure, defining a notion of type for those sequences. We describe the substitutive structure of the smaller subshifts obtained by fixing the sequence of types of the successive derivatives of smooth sequences, from which we obtain the unique ergodicity of all these subshifts. A direct consequence is that the asymptotic frequency of any finite pattern in a smooth sequence over {1,3} is always well-defined and depends on its type sequence. Finally, we characterize the minimality of these subshifts.

cs.DM↗

arXiv2022

Factor-balanced $S$-adic languages

A set of words, also called a language, is letter-balanced if the number of occurrences of each letter only depends on the length of the word, up to a constant. Similarly, a language is factor-balanced if the difference of the number of occurrences of any given factor in words of the same length is bounded. The most prominent example of a letter-balanced but not factor-balanced language is given by the Thue-Morse sequence. We establish connections between the two notions, in particular for languages given by substitutions and, more generally, by sequences of substitutions. We show that the two notions essentially coincide when the sequence of substitutions is proper. For the example of Thue-Morse-Sturmian languages, we give a full characterisation of factor-balancedness.

cs.FL↗