  We study the receptive metric entropy for semigroup actions on probability
spaces, inspired by a similar notion of topological entropy introduced by
Hofmann and Stoyanov. We analyze its basic properties and its relation with the
classical metric entropy. In the case of semigroup actions on compact metric
spaces we compare the receptive metric entropy with the receptive topological
entropy looking for a Variational Principle. With this aim we propose several
characterizations of the receptive topological entropy. Finally we introduce a
receptive local metric entropy inspired by a notion by Bowen generalized in the
classical setting of amenable group actions by Zheng and Chen, and we prove
partial versions of the Brin-Katok Formula and the local Variational Principle.
