For a surface S with n marked points and fixed genus g 2, we prove that the logarithm of the minimal dilatation of a pseudo-Anosov homeomorphism of S is on the order of .log n/=n. This is in contrast with the cases of genus zero or one where the order is 1=n.

We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus g > 2, we show that there are positive constants a 1 < a 2 such that the minimal translation length is bounded below and above by a 1 =g 2 and a 2 =g 2 .