  In this paper we show that the invariant Cantor set of period doubling type
of any infinitely renormalizable area-preserving map in the universality class
of the Eckmann-Koch-Wittwer renormalization fixed point is always contained in
a Lipschitz curve but never contained in a smooth curve. This extends previous
results by de Carvalho, Lyubich and Martens about strongly dissipative maps of
the plane close to unimodal maps to the area-preserving setting. The method
used for constructing the Lipschitz curve is very similar to the method used in
the dissipative case but proving the nonexistence of smooth curves requires new
techniques.
