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 inf(n --> + infinity)p(n)/n(2) less than or equal to 1/2, but give examples with lim sup(n --> + infinity)p(n)/G(n) = 1 for any prescribed function G with G(n) = o(a(n)) 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 greater than or equal to 1, the rank is at most 2[a], with bounded strings of spacers, and the system is generated by a finite number of substitutions.