Rank and symbolic complexity

We investigate the relation between the complexity function of a sequence, that is the number p(n) of its factors of length n, and the rank of the associated dynamical system, that is the number of Rokhlin towers required to approximate it. We prove that if the rank is one, then lim infn→+∞ p(n) n2 ≤ 1 2 , but give examples with lim supn→+∞ p(n) G(n) = 1 for any prescribed function G with G(n) = o(a) for every a > 1. We give exact computations for examples of the ”staircase” type, which are strongly mixing systems with quadratic complexity. Conversely, for minimal sequences, if p(n) < an + b for some a ≥ 1, the rank is at most 2 [a], with bounded strings of spacers, and the system is generated by a finite number of substitutions. Given a dynamical system, there are several notions indicating that it has a simple structure. One is the notion of rank, defined in [ORN-RUD-WEI] to formalize some constructions initiated by [CHA]; it is purely measuretheoretic, but leads to symbolic constructions, with systems defined by sequences on a finite alphabet. Another one, which may be tracked back to [HED-MOR1], is the combinatorial notion of complexity function, of languages or sequences; since the famous works of Hedlund and Morse, it is