Topological mild mixing of all orders along polynomials

Abstract

<p style='text-indent:20px;'>A minimal system <inline-formula><tex-math id="M1">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> is topologically mildly mixing if for all non-empty open subsets <inline-formula><tex-math id="M2">\begin{document}$ U,V $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ \{n\in {\mathbb Z}: U\cap T^{-n}V\neq \emptyset\} $\end{document}</tex-math></inline-formula> is an IP<inline-formula><tex-math id="M4">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-set. In this paper we show that if a minimal system is topologically mildly mixing, then it is mild mixing of all orders along polynomials. That is, suppose that <inline-formula><tex-math id="M5">\begin{document}$ (X,T) $\end{document}</tex-math></inline-formula> is a topologically mildly mixing minimal system, <inline-formula><tex-math id="M6">\begin{document}$ d\in {\mathbb N} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ p_1(n),\ldots, p_d(n) $\end{document}</tex-math></inline-formula> are integral polynomials with no <inline-formula><tex-math id="M8">\begin{document}$ p_i $\end{document}</tex-math></inline-formula> and no <inline-formula><tex-math id="M9">\begin{document}$ p_i-p_j $\end{document}</tex-math></inline-formula> constant, <inline-formula><tex-math id="M10">\begin{document}$ 1\le i\neq j\le d $\end{document}</tex-math></inline-formula>. Then for all non-empty open subsets <inline-formula><tex-math id="M11">\begin{document}$ U , V_1, \ldots, V_d $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ \{n\in {\mathbb Z}: U\cap T^{-p_1(n) }V_1\cap T^{-p_2(n)}V_2\cap \ldots \cap T^{-p_d(n) }V_d \neq \emptyset \} $\end{document}</tex-math></inline-formula> is an IP<inline-formula><tex-math id="M13">\begin{document}$ ^* $\end{document}</tex-math></inline-formula>-set. We also give the corresponding theorem for systems under abelian group actions.</p>