Switched server systems are mathematical models of manufacturing, traffic and queueing systems that have being studied since the early 1990s. In particular, it is known that typically the dynamics of such systems is asymptotically periodic: each orbit of the system converges to one of its finitely many limit cycles. In this article, we provide an explicit example of a switched server system with exotic behaviour: each orbit of the system converges to the same Cantor attractor. To accomplish this goal, we bring together recent advances in the understanding of the topological dynamics of piecewise contractions and interval exchange transformations (IETs) with flips. The ultimate result is a switched server system whose Poincaré map is semiconjugate to a minimal and uniquely ergodic IET with flips.

中文翻译：

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与最小间隔交换半共轭的交换服务器系统

交换服务器系统是自 1990 年代初以来一直在研究的制造、交通和排队系统的数学模型。特别是，众所周知，这种系统的动力学通常是渐近周期性的：系统的每个轨道都收敛到其有限多个极限环之一。在本文中，我们提供了一个具有奇异行为的切换服务器系统的显式示例：系统的每个轨道都收敛到同一个 Cantor 吸引子。为了实现这一目标，我们将最近在理解分段收缩和间隔交换变换 (IET) 的拓扑动力学方面取得的进展与翻转结合在一起。最终的结果是一个切换的服务器系统，其 Poincaré 映射与具有翻转的最小且唯一的遍历 IET 是半共轭的。