Abstract:The semi-dynamical system of a continuous self-map is generated by iteration of the map, however, the iteration itself, being an operator on the space of continuous self-maps, may generate interesting dynamical behaviors. In this paper we prove that the iteration operator is continuous on the space of all continuous self-maps of a compact metric space and therefore induces a semi-dynamical system on the space. Furthermore, we characterize its fixed points and periodic points in the case that the compact metric space is a compact interval by discussing the Babbage equation. We prove that all orbits of the iteration operator are bounded but most fixed points are not stable. On the other hand, we prove that the iteration operator is not chaotic.

---

中文翻译：

---

连续算子空间上迭代算子的动力学

摘要：连续自我图的半动力学系统是由地图的迭代生成的，但是，迭代本身作为连续自我图空间上的算子，可能会产生有趣的动力学行为。在本文中，我们证明了迭代算子在紧凑度量空间的所有连续自映射的空间上是连续的，因此在该空间上诱导出一个半​​动力学系统。此外，通过讨论巴贝奇方程，我们在紧度量空间是紧区间的情况下，描述了它的不动点和周期点。我们证明了迭代算子的所有轨道都是有界的，但是大多数固定点都是不稳定的。另一方面，我们证明了迭代算子不是混沌的。

---