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Abstract

This paper defines measure-theoretic pressure for an amenable group action by using spanning sets, and shows that the measure-theoretic pressure of an ergodic measure can be described in terms of metric entropy and an integral of the observable associated with the ergodic measure. Using the theory of Carathéodory structure, we give an equivalent definition of measure-theoretic pressure for amenable group actions, and obtain an inverse variational principle, i.e., the topological pressure on a certain set is exactly the measure-theoretic pressure of an ergodic measure.