On the general fractal dimensions of hyperspace of compact sets

Abstract

Consider a separable metric space $(X,d)$, and let $\left(K\right(X),\tilde{d})$ denote the space of non-empty compact subsets of X equipped with the Hausdorff metric. This paper aims to introduce and investigate the concepts of two general fractal dimensions and general dimensions within the framework of $\left(K\right(X),\tilde{d})$. In particular, we explore a relationship between the general fractal dimensions of a set Z of a self-similar sequence space and their counterparts in the space of compact subsets $K\left(Z\right)$.