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

TL;DR

This paper investigates the geometric properties of invariant sets in area-preserving maps, revealing that both bounded and unbounded geometries coexist on subsets of positive measure, contrasting previous dissipation-focused results.

Contribution

It demonstrates that for area-preserving Hénon-like maps, invariant sets exhibit both bounded and unbounded geometries on positive measure subsets, extending understanding beyond dissipative cases.

Findings

Existence of both bounded and unbounded geometries in invariant sets.

Positive measure subsets exhibit unbounded geometry.

Contrasts with previous results on dissipative maps.

Abstract

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.