Scaled packing entropy for amenable group actions

In order to characterize the complexity of a system with zero or infinite entropy, we introduce the notions of scaled packing entropies in the framework of countable discrete amenable group actions by describing the speed of divergence of nearby orbits by any scaled sequences. After presenting some basic properties of the scaled packing entropy with respect to the scaled sequences, a variational principle is established. We show that for any Borel subset Z of X, the scaled packing topological entropy of Z equals the supremum of upper local entropy over all Borel probability measures for which the subset Z has full measure. The variational principle for the scaled packing entropy dimension is also established. Finally, the amenable scaled packing entropy on the set of generic points is discussed and inequalities via factor maps are also obtained.