迭代乘积的乘极小点
Multiply minimal points for the product of iterates
摘要：
多重Birkhoff递归定理指出，对于任何$d\in\mathbb N$，每个系统$(X,T)$都有一个多重递归点$x$，即$(x,x,\ldots, x)$在$τ_d=:T\times T^2\times \ldots \times T^d$下是递归的。很自然地会问是否总有一个乘极小点，即一个点$x$使得$(x,x,\ldots,x)$是$τ_d$ -极小的。本文通过对均圈流的研究，给出了否定的答案。然而，它表明，对于任何最小系统$(X,T)$和任何非空开集$U$，有$x\in U$，使$\{n\in{\mathbb Z}: T^nx\in U, \ldots, T^{dn}x\in U\}$是分段syndetic的;和PI最小系统，任何$M$ -子系统的$(X^d, τ_d)$是最小的。
英文摘要：
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 $τ_d=:T\times T^2\times \ldots \times T^d$. 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 $τ_d$-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 $(X^d, τ_d)$ is minimal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
