A New Way to Analyze Iterative Systems through Topology

Ponente(s): Cesar Alfonso Ruiz Alexander

Defining a dynamical system based on iterations of a given function is perhaps the simplest manner of doing so. Problems regarding their long-term behavior remain as elusive as ever, with most results regarding them being induced by an overarching structure that often proves to be too specific to be generalized to all possible constructions. The formalism in question is a topology one can associate to a map defined in terms of a concept we call stable sets. This topology has the peculiarity of relating the long term behavior of the system with its intrinsic, topological properties. Of particular note, we prove that the existence of a non-preperiodic point is equivalent to its associated topology being path connected.  One should be advised that the topologies found by applying this formalism are very quaint, as in general they are irreducible.  We begin by introducing the concept of a stable set and outline a few indispensable properties they enjoy; then, we define the dynamic topology associated to a system and proceed to outline a foray into the property they enjoy in general (over the natural numbers); lastly, we delve into a case study dealing with a set of particularily amicable functions.