Let (X, G) be a G-action topological dynamical system (t.d.s. for short), where G is a countably infinite discrete amenable group. In this paper, we study the topological pressure of the sets of generic points. We show that when the system satisfies the almost specification property, for any G-invariant measure μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} and any continuous map φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document}, PXμ,φ,{Fn}=hμ(X)+∫φdμ,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} P\left( X_{\mu },\varphi ,\{F_n\}\right) = h_{\mu }(X)+\int \varphi d\mu , \end{aligned}$$\end{document}where {Fn}\documentclass[12pt]{minimal}
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\begin{document}$$\{F_n\}$$\end{document} is a Følner sequence, Xμ\documentclass[12pt]{minimal}
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\begin{document}$$X_{\mu }$$\end{document} is the set of generic points of μ\documentclass[12pt]{minimal}
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\begin{document}$$\mu $$\end{document} with respect to (w.r.t. for short) {Fn}\documentclass[12pt]{minimal}
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\begin{document}$$\{F_n\}$$\end{document}, P(Xμ,φ,{Fn})\documentclass[12pt]{minimal}
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\begin{document}$$P(X_{\mu },\varphi ,\{F_n\})$$\end{document} is the topological pressure of Xμ\documentclass[12pt]{minimal}
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\begin{document}$$X_{\mu }$$\end{document} for φ\documentclass[12pt]{minimal}
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\begin{document}$$\varphi $$\end{document} w.r.t. {Fn}\documentclass[12pt]{minimal}
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\begin{document}$$\{F_n\}$$\end{document} and hμ(X)\documentclass[12pt]{minimal}
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\begin{document}$$h_{\mu }(X)$$\end{document} is the measure-theoretic entropy.
