Coexistence of bounded and unbounded geometry for area-preserving maps

Abstract

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: bounded and unbounded geometries coexist with both phenomena occuring on subsets of positive measure in the Cantor sets. Contents 16 6. Bounded geometry 19 References 23