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Conditional Ergodic Averages for Asymptotically Additive Potentials

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.