Hausdorff dimension and infinitesimal similitudes on complete metric spaces

Abstract: In this paper, we answer a question of Nussbaum, Priyadarshi, and Lunel [Positive operators and Hausdorff dimension of invariant sets, Trans. Amer. Math. Soc. 364(2) (2012) 1029-1066.]. We also show that the Hausdorff dimension and box dimension of the attractor generated by a finite set of contractive infinitesimal similitudes are the same. Further, we extend many results of dimension theory to complete metric spaces. In the last part, we fill the gaps in the proofs of some articles, which are related to the dimension theory, and hint at some possible improvements in the papers.