On the topological entropy of the free semigroup action generated by proper maps

In this paper, we introduce the topological entropy of a free semigroup action generated by proper maps, which extends the notions of the topological entropy of the free semigroup actions defined by Bufetov in 1999 and topological entropy of the proper maps defined by Patrão in 2010. We then give some properties of these notions and discuss the relations between them. We also give a partial variational principle for locally compact separable metric spaces. Moreover, the relationship between topological entropy of the free semigroup generated by proper maps and topological entropy of a skew-product transformation is given. These results extend the results obtained by Patrão, Bufetov and Lin, Ma and Wang in 2018.