Topological mild mixing of all orders along polynomials

Abstract

A minimal system $(X,T)$ is topologically mildly mixing if all non-empty open subsets $U,V$, $\{n\in \mathbb{Z}:U\cap T^{-n}V\ne \varnothing \}$ is an IP${}^{\ast }$-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 $(X,T)$ is a topologically mildly mixing minimal system, $d\in \mathbb{N}$, $p_1(n),\dots ,p_d(n)$ are integral polynomials with no $p_i$ and no $p_i-p_j$ constant, $1\le i\ne j\le d$, then for all non-empty open subsets $U,V_1,\dots ,V_d$, $\{n\in \mathbb{Z}:U\cap T^{-p_1(n)}{V}_1\cap T^{-p_2(n)}{V}_2\cap \dots \cap T^{-p_d(n)}{V}_d\ne \varnothing \}$ is an IP${}^{\ast }$-set. We also give the theorem for systems under abelian group actions.