Multiply minimal points for the product of iterates

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. $(x,x,\dots ,x)$ is recurrent under ${\tau }_d=:T\times T^2\times \dots \times T^d$. It is natural to ask if there always is a multiply minimal point, i.e. a point $x$ such that $(x,x,\dots ,x)$ 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 $\{n\in \mathbb{Z}:T^{n}x\in U,\dots ,T^{dn}x\in U\}$ is piecewise syndetic; and that for a PI minimal system, any $M$-subsystem of $(X^d,{\tau }_d)$ is minimal.