Variational principle for neutralized packing pressure on subsets

TL;DR

This paper establishes a variational principle linking neutralized packing pressures and measure-theoretic pressures for subsets under finitely generated free semigroup actions on compact metric spaces.

Contribution

It introduces the concepts of neutralized packing and measure-theoretic pressures and proves a variational principle connecting them for semigroup actions.

Findings

Established the variational principle between neutralized packing and measure-theoretic pressures.

Defined new notions of neutralized pressures for subsets in dynamical systems.

Demonstrated the equality of these pressures in the context of finitely generated free semigroup actions.

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 $\overline{P}_{\mu,{\mathcal{G}} }(Z,f)$ for a given continuous function $f$ and a compact subset $Z \subset X$: $$P^{P}_{\mathcal{G}}(Z,f)=\lim_{\varepsilon \to 0}\sup \{ \overline{P}_{\mu,\mathcal{G} }(Z,f,\varepsilon):\mu \in M(X), \ \mu(Z)=1 \}.$$