We define by simple conditions two wide subclasses of the so-called Arnoux-Rauzy systems; the elements of the first one share the property of (measure-theoretic) weak mixing, thus we generalize and improve a counter-example to the conjecture that these systems are codings of rotations; those of the second one have eigenvalues, which was known hitherto only for a very small set of examples.
Let I be a finite set of words and be the derivation relation generated by the set of productions {ε → u | u ∈ I}. Let be the set of words u such that . We prove that the set I is unavoidable if and only if the relation is a well quasi-order on the set . This result generalizes a theorem of [Ehrenfeucht et al.,Theor. Comput. Sci.27 (1983) 311–332]. Further generalizations are investigated.
Given an ordered alphabet and a permutation, according to the lexicographic order, on the set of suffixes of a word , we present in this article a linear time and space method to determine whether a word has the same permutation on its suffixes. Using this method, we are then also able to build the class of all the words having the same permutation on their suffixes, first of all the smallest one. Finally, we note that this work can lead to a method for generating a Lyndon word randomly in linear...
Given an ordered alphabet and a permutation, according to the lexicographic order, on the set of suffixes of a word w, we present in this article a linear time and space method to determine whether a word w' has the same permutation on its suffixes. Using this method, we are then also able to build the class of all the words having the same permutation on their suffixes, first of all the smallest one. Finally, we note that this work can lead to a method for generating a Lyndon word...