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The upper bounds of Friedland’s entropy for certain -actions

In this paper, we study Friedland’s entropy for -actions, which was introduced by Friedland in 1996. Unlike the traditional definition, it is non-zero for actions whose generators have finite entropy as single transformations. We investigate two types of -actions: (1) -actions on finite graphs, and (2) expanding -actions on closed Riemannian manifolds, and show that their preimage branch entropies, which rely on the preimage structure of the actions, are zero. Hence, the upper bounds of Friedland’s entropy for these -actions are obtained.