Variational principle for neutralized packing pressure on subsets

In this paper, we introduce the notions of neutralized packing pressures and neutralized measure-theoretic pressures on subsets for a finitely generated free semigroup action. Let $X$ be a compact metric space and $\mathcal{G}$ be a finite family of continuous self-maps on $X$. We consider the semigroup $G$ generated by $\mathcal{G}$ on $X$. We show that the variational principle between the neutralized packing pressures $P_{\mathcal{G}}^P(Z,f)$ and the neutralized measure–theoretic upper pressures ${\bar{P}}_{\mu ,\mathcal{G}}(Z,f)$ for a given continuous function $f$ and a compact subset $Z\subset X$: $$P_{\mathcal{G}}^P(Z,f)=\underset{\epsilon \to 0}{\lim }\sup \{{\bar{P}}_{\mu ,\mathcal{G}}(Z,f,\epsilon ):\mu \in M(X),\mu (Z)=1\}\mathrm{.}$$