Multiply minimal points for the product of iterates

The multiple Birkhoff recurrence theorem states that for any $dinmathbb N$, every system $(X,T)$ has a multiply recurrent point $x$, i.e. $(x,x,ldots, x)$ is recurrent under $tau_d=:Ttimes T^2times ldots times T^d$. 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 $xin U$ such that ${nin{mathbb Z}: T^nxin U, ldots, T^{dn}xin U}$ is piecewise syndetic; and that for a PI minimal system, any $M$-subsystem of $(X^d, tau_d)$ is minimal.