  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.
