Topological Entropy of Free Semigroup Actions Generated by Proper Maps for Noncompact Subsets

In this paper, we introduce three notions of topological entropy a free semigroup action generated by proper maps for noncompact subsets, which extends the defined Ju et al. [13] and Ma [17]. By using one-point compactification as bridge, study relations entropies between two dynamical systems. We then skew-product transformations, particular subset, relationship upper capacity maps, transformation is given. As applications, examine multifractal spectrum locally compact separable metric space, it shown that irregular set has full maps.