On the geometry of period doubling invariant sets for area-preserving maps

The geometry of the period doubling Cantor sets of strongly dissipative infinitely renormalizable Hénon-like maps has been shown to be unbounded by M. Lyubich, M. Martens, and A. de Carvalho, although the measure of unbounded 'spots' in the Cantor set has been demonstrated to be zero.We show that an even more extreme situation takes places for infinitely renormalizable area-preserving Hénon-like maps: Both bounded and unbounded geometries exist on subsets of positive measure.