Multiply minimal points for the product of iterates

TL;DR

This paper investigates the existence of multiply minimal points in dynamical systems, showing they do not always exist, but establishing conditions under which certain recurrence properties hold, especially in minimal and PI systems.

Contribution

It demonstrates the non-existence of multiply minimal points in general, and proves recurrence properties for minimal and PI systems, advancing understanding of multiple recurrence phenomena.

Findings

No always multiply minimal points in general systems.

Existence of points with piecewise syndetic recurrence in minimal systems.

Minimal subsystems of PI systems are minimal under product maps.

Abstract

The multiple Birkhoff recurrence theorem states that for any $d\in\mathbb N$, every system $(X,T)$ has a multiply recurrent point $x$, i.e. $(x,x,\ldots, x)$ is recurrent under $\tau_d=:T\times T^2\times \ldots \times T^d$. It is natural to ask if there always is a multiply minimal point, i.e. a point $x$ such that $(x,x,\ldots,x)$ is $\tau_d$-minimal. A negative answer is presented in this paper via studying the horocycle flows. However, it is shown that for any minimal system $(X,T)$ and any non-empty open set $U$, there is $x\in U$ such that $\{n\in{\mathbb Z}: T^nx\in U, \ldots, T^{dn}x\in U\}$ is piecewise syndetic; and that for a PI minimal system, any $M$-subsystem of $(X^d, \tau_d)$ is minimal.