On the Iterative Behavior of Open Maps

拓扑空间之间的开映射已被广泛研究多年。例如，Stoilow在1900年初表明，2球的每个开放和光照地图都是一个分支覆盖地图。然而，人们对这种映射的迭代行为知之甚少。例如，球的每一个分支覆盖映射是否在每一个具有连通补的性质不变子连续域中都有一个不动点仍然是未知的。相比之下，使用大多数分析技术，球体的有理映射的动力学被很好地理解。然而，在某些极限情况下（即，如果R有中性周期点或循环临界点），解析方法是困难的。在对所有连续函数的动力学进行详细分析方面取得重大进展的唯一情况是单位区间的映射。我们相信，对于其他简单空间(如：（流形或树突），没有分析或一对一的假设。在这个方向上取得了一些进展，包括流形上的最小映射（不一定是一对一的）和球面上的开放和单调映射的某些组合。系统的长期行为是由一组方程和给定的初始条件决定的。系统可以用作用于空间X的函数f来描述，函数f具有给定的初始值X（“系统状态”）。然后，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时的状态导致值x(1)，时间1时的状态导致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。系统的复杂性由空间X和函数f的复杂性决定。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关于系统长期行为的信息。

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.