Entropies of compact subsets and supported measures

Let $(X,T)$ be a topological dynamical system and $(\mathcal{M}(X),T_{\ast })$ be its induced system. For a non-empty compact subset $K\subset X$, we define $\mathcal{M}(K)$ as the set of Borel probability measures supported on $K$. In this paper, we systematically study the relationship between various entropies of $(T,K)$ and of $(T_{\ast },\mathcal{M}(K))$. We show that: \begin{equation*} \begin{aligned} & h_{$\mathrm{t}\mathrm{o}\mathrm{p}$}^{$\mathrm{U}\mathrm{C}$}(T,K)>0 \iff h_{$\mathrm{t}\mathrm{o}\mathrm{p}$}^{$\mathrm{U}\mathrm{C}$}(T_,$\mathcal{M}$(K))=\infty, \qquad &h_{$\mathrm{t}\mathrm{o}\mathrm{p}$}^{P}(T,K)>0\iff h_{$\mathrm{t}\mathrm{o}\mathrm{p}$}^{P}(T_,$\mathcal{M}$(K))>0, \qquad &h_{$\mathrm{t}\mathrm{o}\mathrm{p}$}^{B}(T,K)>0 \implies h_{$\mathrm{t}\mathrm{o}\mathrm{p}$}^{B}(T_,$\mathcal{M}$(K))>0 , \end{aligned} \end{equation} where $h_{\mathrm{t}\mathrm{o}\mathrm{p}}^{\mathrm{U}\mathrm{C}}(T,K)$, $h_{\mathrm{t}\mathrm{o}\mathrm{p}}^P(T,K)$, and $h_{\mathrm{t}\mathrm{o}\mathrm{p}}^B(T,K)$ denote the upper capacity topological entropy, the packing topological entropy, and the Bowen topological entropy of $K$, respectively. Additionally, we present a counterexample involving a non-invariant set, demonstrating that the converse of the third assertion is not valid in general.