  Using an asymptotically additive sequence of continuous functions as a
restrictive condition, this paper studies the relations of several ergodic
averages for asymptotically additive potentials. Basic properties of
conditional maximum ergodic averages are studied. In particular, if the
dynamical systems satisfy the specification property, the maximal growth rate
of an asymptotically additive potential on the level set is equal to its
conditional maximum ergodic averages and the maximal growth rates on the
irregular set is its maximum ergodic averages. Finally, the applications for
suspension flows are given in the end of the paper.
