Complexity of sequences and dynamical systems

In recent years, there has been a number of papers about the combinatorial notion of symbolic complexity: this is the function counting the number of factors of length n for a sequence. The complexity is an indication of the degree of randomness of the sequence: a periodic sequence has a bounded complexity, the expansion of a normal number has an exponential complexity. For a given sequence, the complexity function is generally not of easy access, and it is a rich and instructive work to compute it; a survey of this kind of results can be found in [ALL]. We are interested here in further results in the theory of symbolic complexity, somewhat beyond the simple question of computing the complexity of various sequences. These lie mainly in two directions; first, we give a survey of an open question which is still very much in progress, namely: to determine which functions can be the symbolic complexity function of a sequence. Then, we investigate the links between the complexity of a sequence and its associated dynamical system, and insist on the cases where the knowledge of the complexity function allows us to know either the sequence, or at least the system. This leads to another vast open question, the S-adic conjecture, and to a conceptual (though still conjectural) link with the notion of KolmogorovChaitin complexity for infinite sequences (see section 6). Also, these links with dynamical systems have been of considerable help to ergodic theory,