Abstract
Motivated by a simple iteration on polygons, we study a projectively natural flow on the space of diffeomorphisms of the circle. This flow, which is given by a nonlinear fourth order PDE, has long term existence an uniqueness, and evolves an arbitrary diffeomorphism into a projective transformation. The flow can have radically different behavior, e.g. finite time blow-ups, when it is defined relative to different projective structures on the circle.
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