Let \((X,T)\) be a topological dynamical system. The following are the main results in this paper. Theorem 1. The following conclusions hold: i)\ For an increasing sequence \(S=\{s_1< s_2< ...\}\) of positive integers, \(h_{\mathrm{top}}^S(X,T)=0\) if and only if \(h_{\mathrm{top}}^S(M(X),T)=0\) ii)\ \(\overline{D}(X,T)=\overline{D}(M(X),T)\). Theorem 2. Let \(\mu\in M(X,T)\) be ergodic and \(Q(\mu)\) be the collection of all factors in \(\mu\). Then i)\ For an increasing sequence \(S=\{s_1< s_2< ...\}\) of positive integers, \(h_\mu^S(X,T)=0\) if and only if \(\sup\{h_\eta^S(M(X),T); \eta\in Q(\mu)\}=0\) ii)\ \(\sup\{\overline{D}_\eta(M(X),T);\eta\in Q(\mu)\}=\overline{D}_\mu(X,T)\). Further aspects involving the connection with some existing statements in the area are also discussed.