Variational principle of higher dimension weighted pressure for amenable group actions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914834767937536 |
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| author | Yin, Zhengyu Xiao, Zubiao |
| author_facet | Yin, Zhengyu Xiao, Zubiao |
| contents | 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 $π_i:(X_i,G)\to (X_{i+1},G)$ 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_{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 $μ_i=π_{i-1}\circ\cdots\circπ_{1}μ$ is the induced $G$-invariant measure on $X_{i}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04224 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Variational principle of higher dimension weighted pressure for amenable group actions Yin, Zhengyu Xiao, Zubiao Dynamical Systems 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 $π_i:(X_i,G)\to (X_{i+1},G)$ 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_{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 $μ_i=π_{i-1}\circ\cdots\circπ_{1}μ$ is the induced $G$-invariant measure on $X_{i}$. |
| title | Variational principle of higher dimension weighted pressure for amenable group actions |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2310.04224 |