  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.
