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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 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.