  The geometry of the period doubling Cantor sets of strongly dissipative
infinitely renormalizable H\'enon-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 H\'enon-like maps: both bounded and unbounded
geometries exist on subsets of positive measure.
