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Bifix codes, Combinatorics on Words and Symbolic Dynamical Systems

Abstract : Sets of words of linear complexity play an important role in combinatorics on words and symbolic dynamics. This family of sets includes set of factors of Sturmian and Arnoux-Rauzy words, interval exchange sets and primitive morphic sets, that is, sets of factors of fixed points of primitive morphisms. The leading issue of this thesis is the study of minimal dynamical systems, also defined equivalently as uniformly recurrent sets of words. As a main result, we consider a natural hierarchy of minimal systems containing neutral sets, tree sets and specular sets. Moreover, we connect the minimal systems to the free group using the notions of return words and basis of subroups of finite index. Symbolic dynamical systems arising from interval exchanges and linear involutions provide us geometrical examples of this kind of sets. One of the main tool used here is the study of possible extensions of a word in a set, that allows us to determine properties such as the factor complexity. In this manuscript we define the extension graph, an undirected graph associated to each word w in a set S which describes the possible extensions of w in S on the left and the right. In this thesis we present several classes of sets of words defined by the possible shapes that the graphs of elements in the set can have. One of the weakest condition that we will study is the neutrality condition: a word w is neutral if the number of pairs (a, b) of letters such that awb ∈ S is equal to the number of letters a such that aw ∈ S plus the number of letters b such that wb ∈ S minus 1. A set such that every nonempty word satisfies the neutrality condition is called a neutral set. A stronger condition is the tree condition: a word w satisfies this condition if its extension graph is both acyclic and connected. A set is called a tree set if any nonempty word satisfies this condition. The family of recurrent tree sets appears as a the natural closure of two known families, namely the Arnoux-Rauzy sets and the interval exchange sets. We also introduce specular sets, a remarkable subfamily of the tree sets. These are subsets of groups which form a natural generalization of free groups. These sets of words are an abstract generalization of the natural codings of interval exchanges and of linear involutions. For each class of sets considered in this thesis, we prove several results concerning closure properties (under maximal bifix decoding or under taking derived words), cardinality of the bifix codes and set of return words in these sets, connection between return words and basis of the free groups, as well as between bifix codes and subgroup of the free group. Each of these results is proved under the weakest possible assumptions
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Francesco Dolce. Bifix codes, Combinatorics on Words and Symbolic Dynamical Systems. Computation and Language [cs.CL]. Université Paris-Est, 2016. English. ⟨NNT : 2016PESC1036⟩. ⟨tel-01532193⟩



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