Open maps between topological spaces have been studied extensively for many years. For example, Stoilow showed in early 1900 that each open and light map of the 2-sphere is a branched covering map. However, much less is known about the iterative behavior of such maps. For example it is still not known if every branched covering map of the sphere has a fixed point in every proper invariant sub-continuum with connected complement. In contrast, using mostly analytic techniques, the dynamics of rational maps R of the sphere are quite well understood. However, the analytic approach is difficult in certain limit cases (i.e., if R has either neutral periodic points or recurrent critical points). The only case where significant progress was made towards a detailed analysis of the dynamics of all continuous functions is for maps of the unit interval. We believe that significant new results are also possible for open (and open-like) maps on other simple spaces (i.e., manifolds or dendrites) without an analytic or one-to-one assumption. Some progress in this direction was made for minimal (not necessarily one-to-one) maps on manifolds and certain compositions of open and monotone maps on the sphere.
The long term behavior of a system is determined by a set of equations and given initial conditions. The system can be described by a function f, acting on a space X, with a given initial value x (the "state of the system"). The long term behavior of the system can then be described by iteration: the initial value x=x(0), the state of the system at time zero, leads to 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 the long term behavior of the system. It is known that, even for very simple systems, the long term behavior of the system may be chaotic. The complexity of the system is governed by the complexity of the space X and the function f. In this research project we propose to extend information about the long term behavior of systems using more general spaces and a larger class of functions.
