Topological mild mixing of all orders along polynomials

TL;DR

This paper proves that minimal systems with topological mild mixing exhibit mild mixing of all polynomial orders, extending the concept to systems with abelian group actions, and characterizes their recurrence properties.

Contribution

It establishes that topologically mildly mixing minimal systems are also mildly mixing of all polynomial orders, including systems with abelian group actions, broadening the scope of mixing properties.

Findings

Topologically mildly mixing systems are mildly mixing of all polynomial orders.

Recurrence sets in such systems are IP*-sets for polynomial iterates.

Extension of results to systems under abelian group actions.

Abstract

A minimal system $(X,T)$ is topologically mildly mixing if all non-empty open subsets $U,V$, $\{n\in \Z: U\cap T^{-n}V\neq \emptyset\}$ is an IP$^*$-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, $d\in \N$, $p_1(n),\ldots, p_d(n)$ are integral polynomials with no $p_i$ and no $p_i-p_j$ constant, $1\le i\neq j\le d$, then for all non-empty open subsets $U , V_1, \ldots, V_d $, $\{n\in \Z: U\cap T^{-p_1(n) }V_1\cap T^{-p_2(n)}V_2\cap \ldots \cap T^{-p_d(n) }V_d \neq \emptyset \}$ is an IP$^*$-set. We also give the theorem for systems under abelian group actions.