Variational principle of higher dimension weighted pressure for amenable group actions

Let $r\geq 2$ 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 $0\leq w_i\leq 1$. In this article, for $f\in C(X_1)$, we introduce the weighted topological pressure $P^{\textbf{a}}(f,G)$ 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^{\textbf{a}}(f,G)=\sup_{\mu\in \mathcal{M}^G(X_1)}\Big(\sum_{i=1}^rw_ih_{\mu_i}(X_i,G)+w_1\int_{X_1}fd\mu\Big), \end{align*} where $\mu_i=\pi_{i-1}\circ\cdots\circ\pi_{1}\mu$ is the induced $G$-invariant measure on $X_{i}$.