The period-doubling Cantor sets of strongly dissipative Hénon-like maps with different average Jacobian are not smoothly conjugated, as was shown previously. The Jacobian rigidity conjecture says that the period-doubling Cantor sets of two-dimensional Hénon-like maps with the same average Jacobian are smoothly conjugated. This conjecture is true for average Jacobian zero, for example, the one-dimensional case. The other extreme case is when the maps preserve area, for example, when the average Jacobian is one. Indeed, the main result presented here is that the period-doubling Cantor sets of area-preserving maps in the universality class of the Eckmann–Koch–Wittwer renormalization fixed point are smoothly conjugated.
"Rigidity for infinitely renormalizable area-preserving maps." Duke Math. J. 165 (1) 129 - 159, 15 January 2016. https://doi.org/10.1215/00127094-3165327