Topological formulation of termination properties of iterates of functions

Abstract

We consider a number of decision problems, that appear in the dynamical systems and database literature, concerning the termination of iterates of real functions. These decision problems take a function f :Rn →Rn as input and ask, for example, whether this function is mortal, nilpotent, terminating, or reaches a fixed point on a given point in Rn.We associate topologies to functions f :Rn→Rn and study some basic properties of these topologies. The contribution of this paper is a translation of the above mentioned decision problems into decision problems concerning well-known properties of topologies, e.g., connectivity. We also show that connectivity of topologies on Rn is undecidable for n>1.