Jean-Pierre Borel

On Christoffel classes

We characterize conjugation classes of Christoffel words (equivalently of standard words) by the number of factors. We give several geometric proofs of classical results on these words and sturmian words.

Palindromic factors of billiard words

We study palindromic factors of billiard words, in any dimension. There are differences between the two-dimensional case, and higher dimension. Arbitrary long palindrome factors exist in any dimension, but arbitrary long palindromic prefixes exist in general only in dimension 2.

Self-similar measures and sequences

Abstract We study asymptotical properties of some particular sequences U = ( u n ) n≥1 of real numbers 0≤ u n ≤1. These sequences are called “self-similar”: if we only consider the restriction of the entire sequence U to (correctly chosen) small sub-intervals I of [0, 1], we obtain the image φ( U ) of U by some regular bijection φ: [0, 1] → I . General construction of such sequences is given, and necessary and sufficient conditions for the existence of an asymptotic distribution of U in [0, 1]. These distributions are obtained as invariant points of some particular transformations T on the setPof all probability measures on [0, 1]. Such transformations are studied, in relation with general methods of representation of real numbers. Some ergodic properties of the shift are obtained in these cases, and the nature of the T -invariant measures is discussed. We also give, in the case of uniform distribution of U , an explicit formula for the error terms. This implies a majoration of the discrepancy, related to some hypothesis on U .

How to build billiard words using decimations

We present two methods based on decimation for computing finite billiard words on any finite alphabet. The first method computes finite billiard words by iteration of some transformation on words. The number of iterations is explicitly bounded. The second one gives a direct formula for the billiard words. Some results remain true for infinite standard Sturmian words, but cannot be used for computation as they only are limit results.

Suites de longueur minimale associees a un ensemble normal donne

For a given subsetA of the set of real numbers, we search a sequence Λ=(λn) of real numbers such that bothA is the normal setB(Λ) associated to Λ, and Λ takes its values in a bounded interval, with a minimal lengthM. A lower bound ofM is obtained, which gives some necessary conditions of existency of such a bounded sequence Λ. More details are given whenA is a subset of the set of integers. In this case, the problem is to find a polynomialQ of lowest degree such that the productP.Q has non-negative coefficients, for some special given polynomialP.

Complexity of degenerated three dimensional billiard words

We consider Billiard Words on alphabets with k = 3 letters: such words are associated to some 3-dimensional positive vector

Some new results on palindromic factors of billiard words

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Opérations sur les mots de Christoffel

. The language of these words is already known in the usual case, i.e., when the αj are linearly independent over

Symbolic representation of piecewise linear functions on the unit interval and application to discrepancy

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Translation of a Digital Line into another according to various Digitization Processes

, and so for the