Classical local rigidity KAM result says that a small perturbation of a Diophantine flow on the torus is, after a small constant modification, smoothly conjugate to the Diophantine flow. I will explain that for higher dimensional "flows" the natural corresponding notion is "transversal" local rigidity. This is proved so far for two classes of homogeneous parabolic R^2 actions, and I will discuss these examples.
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