Variational principle of higher dimension weighted pressure for amenable group actions

Let $r\ge 2$ and $(X_i,G)$ $(i=1,\cdots ,r)$ be topological dynamical systems with $G$ being an infinite discrete amenable group. Suppose that ${\pi }_i:(X_i,G)\to (X_{i+1},G)$ are factor maps and $0\le w_i\le 1$. In this article, for $f\in C(X_1)$, we introduce the weighted topological pressure $P^{\mathbf{\text{a}}}(f,G)$ for higher dimensions (not only for $r=2$) of amenable group actions. By using measure-theoretical theory, we establish a variational principle as \begin{align*} P^{$\mathbf{\text{a}}$}(f,G)=\sup_{μ\in $\mathcal{M}$^G(X_1)}\Big(\sum_{i=1}^rw_ih_{μ_i}(X_i,G)+w_1\int_{X_1}fdμ\Big), \end{align*} where ${\mu }_i={\pi }_{i-1}\circ \cdots \circ {\pi }_1\mu$ is the induced $G$-invariant measure on $X_i$.