Conditional Ergodic Averages for Asymptotically Additive Potentials

TL;DR

This paper investigates the properties of ergodic averages for asymptotically additive potentials, establishing their behavior under specification conditions and exploring applications to suspension flows.

Contribution

It introduces a framework for analyzing conditional maximum ergodic averages for asymptotically additive potentials, extending understanding of their growth rates in dynamical systems.

Findings

Maximal growth rate equals conditional maximum ergodic averages under specification.

Maximal growth rates on irregular sets match maximum ergodic averages.

Applications demonstrated for suspension flows.

Abstract

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.