We show that bounded measurable and non-degenerate vector fields u on RN can be written as a composition u = R of a maximal monotone operator and an “essentially idempotent” measure preserving map R. The vector field is not cyclically monotone (i.e., of the form r with being a convex potential) as in Brenier’s celebrated polar decomposition, however the measure preserving factor R in our decomposition is better behaved since it is essentially an idempotent map (i.e., R2 = I). The proof relies on a new version of optimal transport. This is a joint work with Abbas Moameni.
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