标题： 轨道扩张映射 显示英文标题

标题： Orbitwise expansive maps

摘要： 本研究定义了一个轨道逐点扩张点（OE），即如果在一个度量空间$(X,\rho)$中存在一个点$x$，并且存在一个数$d>0$，使得任意开球内几个点的轨道至少有一次与$x$的轨道上对应点的距离大于$d$，则称该点为轨道逐点扩张点。 点$x$在上述场景中被称为相对轨道膨胀点（ROE），如果将$d$替换为所研究轨道的开球半径，且该球心为$x$。我们考虑生成轨道的函数是连续的。 我们还定义了OE（ROE）集。 我们证明任意OE（ROE）集的并集仍然是OE（ROE）集，并且OE集的每个极限点都是一个OE点。 我们表明，与相反的情况相比，Utz的膨胀映射或Kato的CW-膨胀映射意味着OE（ROE）映射。 我们利用OE（ROE）的概念来分析一个时变的动力系统，并探讨它与某些与膨胀性相关的特征的相关性。 显示英文摘要

摘要： This study defines an orbitwise expansive point (OE) as a point, such as $x$ in a metric space $(X,\rho)$, if there is a number $d>0$ such that the orbits of a few points inside an arbitrary open sphere will maintain a distance greater than $d$ from the corresponding points of the orbit of $x$ at least once. The point $x$ is referred to as the relatively orbitwise expansive point (ROE) in the previously described scenario if $d$ is replaced with the radius of the open sphere whose orbit is investigated and whose centre is $x$. %The function generating the orbit is considered to be continuous. We also define OE (ROE) set. We prove that arbitrary union of OE (ROE) set is again OE (ROE) set and every limit point of an OE set is an OE point. We show that, rather than the other way around, Utz's expansive map or Kato's CW-expansive map implies OE (ROE) map. We utilise the concept of OE(ROE) to analyse a time-varying dynamical system and investigate its relevance to certain traits associated with expansiveness.