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.