On the topological entropy of a semigroup of continuous maps

We introduce the notion of the lower local entropy of a Borel probability measure on a compact metric space to estimate the bounds of topological entropy with respect to any subset in the case of finitely generated semigroup of continuous maps. Moreover, we give the notion of the topological entropy of a semigroup generated by finite uniformly continuous maps on a metric space not necessarily compact, provide some properties of this topological entropy, and estimate the bounds of them for some particular systems, such as a semigroup generated by finite affine transformations on the p-dimensional torus and a semigroup generated by finite smooth maps on Riemannian manifolds.