  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 F are carried by some iterate of $F^{-1}$ to a
class of highly singular inputs of $F^{-1}$. 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.
