In this paper, we introduce three notions of topological entropy of a free semigroup action generated by proper maps for noncompact subsets, which extends the notions defined by Ju et al.  and Ma et al. . By using the one-point compactification as a bridge, we study the relations of the entropies between two dynamical systems. We then introduce three skew-product transformations, and for a particular subset, the relationship between the upper capacity topological entropy of a free semigroup action generated by proper maps, and the upper capacity topological entropy of a skew-product transformation is given. As applications, we examine the multifractal spectrum of a locally compact separable metric space, and it is shown that the irregular set has full upper capacity topological entropy of a free semigroup action generated by proper maps.
The work was supported by National Natural Science Foundation
of China (grant no. 11771149).
The authors really appreciate the editor and referees' valuable remarks and suggestions that helped a lot.
"Topological Entropy of Free Semigroup Actions Generated by Proper Maps for Noncompact Subsets." Taiwanese J. Math. Advance Publication 1 - 24, 2021. https://doi.org/10.11650/tjm/210903