In the context of symbolic dynamics, the class of « sublinear complexity subshifts » is of particular relevance as it occurs in a variety of areas, such as geometric dynamical systems, language theory, number theory, and numeration systems, among others. During the intensive study carried on this subject from the beginning the 90’s, it was proposed that a hierarchical decomposition based on S-adic sequences that characterizes sublinear complexity subshifts would be useful to understand this class. The problem of finding such a characterization was given the name « S-adic conjecture » and inspired several influential results in symbolic dynamics. In this talk, I will present an S-adic characterization of sublinear complexity subshifts and some of its applications, which in particular give a solution to this conjecture.
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