Variational principle for neutralized packing pressure on subsets
Zubiao Xiao, Hongwei Jia
TL;DR
This work introduces neutralized packing pressure and neutralized measure-theoretic upper pressure for finitely generated free semigroup actions on compact spaces and proves a variational principle connecting them. The authors define neutralized Bowen balls, neutralized pressures, and a Katok-style neutralized packing framework, then establish a lower bound via a neutralized Katok construction and an upper bound via a limiting-measure argument. The resulting principle expresses the neutralized topological pressure on a subset as the limit of supremal neutralized measure-theoretic pressures over measures concentrating on the subset. The findings extend Bowen-type and packing-pressure theories to non-amenable semigroup actions and provide a robust tool for analyzing dynamical complexity under neutralized scaling.
Abstract
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 $P^{P}_{\mathcal{G}}(Z,f)$ and the neutralized measure--theoretic upper pressures $\overline{P}_{μ,{\mathcal{G}} }(Z,f)$ for a given continuous function $f$ and a compact subset $Z \subset X$: $$P^{P}_{\mathcal{G}}(Z,f)=\lim_{\varepsilon \to 0}\sup \{ \overline{P}_{μ,\mathcal{G} }(Z,f,\varepsilon):μ\in M(X), \ μ(Z)=1 \}.$$