On the Iterative Behavior of Open Maps

拓扑空间之间的开映射已经被广泛研究了很多年。例如，斯托洛在1900年早期证明了2-球面的每个开放和光照贴图都是一个分支覆盖贴图。然而，对这种映射的迭代行为知之甚少。例如，球面上的每个分支覆盖映射是否在每个具有连通补的真不变子连续统中有不动点，目前还不清楚.相比之下，使用大多数分析技术，球的有理映射R的动力学是相当好理解的。然而，分析方法在某些极限情况下是困难的（即，如果R具有中立周期点或常返临界点）。唯一的情况下，取得了重大进展的详细分析动态的所有连续函数是映射的单位区间。我们相信，对于其他简单空间上的开（和类开）映射（即，流形或枝晶）而没有解析或一对一的假设。在这个方向上，流形上的极小（不一定是一对一）映射和球面上的开映射和单调映射的某些合成都取得了一些进展，系统的长期行为由一组方程和给定的初始条件决定。这个系统可以用一个作用在空间X上的函数f来描述，它有一个给定的初始值x（系统的状态）。系统的长期行为可以通过迭代来描述：初始值x=x（0），系统在时间0的状态，导致值x（1），时间1的状态，导致x（2）等。序列x（0），x（1）的极限行为描述了系统的长期行为。众所周知，即使对于非常简单的系统，系统的长期行为也可能是混沌的。系统的复杂性取决于空间X和函数f的复杂性。在这个研究项目中，我们建议使用更一般的空间和更大的函数类来扩展系统的长期行为信息.

Open maps between topological spaces have been studied extensivelyfor many years. For example, Stoilow showed in early 1900 thateach open and light map of the 2-sphere is a branched coveringmap. However, much less is known about the iterative behavior ofsuch maps. For example it is still not known if every branchedcovering map of the sphere has a fixed point in every properinvariant sub-continuum with connected complement. In contrast,using mostly analytic techniques, the dynamics of rational maps Rof the sphere are quite well understood. However, the analyticapproach is difficult in certain limit cases (i.e., if R haseither neutral periodic points or recurrent critical points). Theonly case where significant progress was made towards a detailedanalysis of the dynamics of all continuous functions is for mapsof the unit interval. We believe that significant new results arealso possible for open (and open-like) maps on other simple spaces(i.e., manifolds or dendrites) without an analytic or one-to-oneassumption. Some progress in this direction was made for minimal(not necessarily one-to-one) maps on manifolds and certaincompositions of open and monotone maps on the sphere.The long term behavior of a system is determined by a set ofequations and given initial conditions. The system can bedescribed by a function f, acting on a space X, with a giveninitial value x (the "state of the system"). The long termbehavior of the system can then be described by iteration: theinitial value x=x(0), the state of the system at time zero, leadsto the value x(1), the state at time one, which leads to x(2)etc. The limit behavior of the sequence x(0), x(1) describes thelong term behavior of the system. It is known that, even for verysimple systems, the long term behavior of the system may bechaotic. The complexity of the system is governed by thecomplexity of the space X and the function f. In this researchproject we propose to extend information about the long termbehavior of systems using more general spaces and a larger classof functions.