Entropy dimensions of the hyperspace of compact sets

Abstract

Let \((X,\rho)\) be a metric space, let \((\mathcal{K}(X),\widetilde{\rho})\) denote the space of non-empty compact subsets of \(X\) with the Hausdorff metric, and let \(E \subset X\). The purpose of this paper is to investigate the relationships between the entropy dimensions of \(E\) and of \(\mathcal{K}(E)\).