We introduce block maps for subfactors and study their dynamic systems. We prove that the limit points of the dynamic system are positive multiples of biprojections and zero. For the Z2 case, the asymptotic phenomenon of the block map coincides with that of the 2 D Ising model. The study of block maps requires a further development of our recent work on the Fourier analysis of subfactors. We generalize the notion of sum set estimates in additive combinatorics for subfactors and prove the exact inverse sum set theorem. Using this new method, we characterize the extremal pairs of Young’s inequality for subfactors, as well as the extremal operators of the Hausdorff-Young inequality.
