Rigidity for infinitely renormalizable area-preserving maps

The period doubling Cantor sets of strongly dissipative Henon-like maps with different average Jacobian are not smoothly conjugated. The Jacobian Rigidity Conjecture says that the period doubling Cantor sets of two-dimensional Henon-like maps with the same average Jacobian are smoothly conjugated. This conjecture is true for average Jacobian zero, e.g. the one-dimensional case. The other extreme case is when the maps preserve area, e.g. the average Jacobian is one. Indeed, the period doubling Cantor set of area-preserving maps in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point are smoothly conjugated.