A variational principle for weighted topological pressure under -actions

Abstract

AbstractLet$k\geq 2$and$(X_{i}, \mathcal {T}_{i}), i=1,\ldots ,k$, be$\mathbb {Z}^{d}$-actions topological dynamical systems with$\mathcal {T}_i:=\{T_i^{\textbf {g}}:X_i{\rightarrow } X_i\}_{\textbf {g}\in \mathbb {Z}^{d}}$, where$d\in \mathbb {N}$and$f\in C(X_{1})$. Assume that for each$1\leq i\leq k-1$,$(X_{i+1}, \mathcal {T}_{i+1})$is a factor of$(X_{i}, \mathcal {T}_{i})$. In this paper, we introduce the weighted topological pressure$P^{\textbf {a}}(\mathcal {T}_{1},f)$and weighted measure-theoretic entropy$h_{\mu }^{\textbf {a}}(\mathcal {T}_{1})$for$\mathbb {Z}^{d}$-actions, and establish a weighted variational principle as$$ \begin{align*} P^{\textbf{a}}(\mathcal{T}_{1},f)=\sup\bigg\{h_{\mu}^{\textbf{a}}(\mathcal{T}_{1})+\int_{X_{1}}f\,d\mu:\mu\in\mathcal{M}(X_{1}, \mathcal{T}_{1})\bigg\}. \end{align*} $$This result not only generalizes some well-known variational principles about topological pressure for compact or non-compact sets, but also improves the variational principle for weighted topological pressure in [16] from$\mathbb {Z}_{+}$-action topological dynamical systems to$\mathbb {Z}^{d}$-actions topological dynamical systems.