Two singularity phenomena

I will discuss two properties that may arise in the study of the singularities of a discrete dynamical system. The first is singularity confinement, introduced by Grammaticos, Ramani, and Papageorgiou. A singularity is said to be confined if it can be bypassed by a higher iterate of the map. This property is well-studied and believed to be closely tied to integrability.
The second property arises from trying to move away from the singularity by applying the inverse map. It is surprisingly common that after a large but predictable number of steps another singularity, mirroring the first, is reached. This behavior was observed by R. Schwartz for the pentagram map. I will demonstrate numerous other examples including Dodgson condensation, Adler's polygon recutting, and some new geometrically-defined systems.