The Devron property

We introduce a criterion called the Devron property that a discrete dynamical system can possess. The Devron property is said to occur when a class of highly singular inputs of a mapping FF are carried by some iterate of F−1F−1 to a class of highly singular inputs of F−1F−1. The inspiration for this definition is the discovery by R. Schwartz that the pentagram map exhibits this kind of behavior. We investigate occurrences of the Devron property in a number of different dynamical systems.