No elliptic islands for the universal area-preserving map
Abstract
A renormalization approach has been used in EKW1 and EKW2 to prove the existence of a universal area-preserving map, a map with hyperbolic orbits of all binary periods. The existence of a horseshoe, with positive Hausdorff dimension, in its domain was demonstrated in GJ1. In this paper the coexistence problem is studied, and a computer-aided proof is given that no elliptic islands with period less than 20 exist in the domain. It is also shown that less than 1.5% of the measure of the domain consists of elliptic islands. This is proven by showing that the measure of initial conditions that escape to infinity is at least 98.5% of the measure of the domain, and we conjecture that the escaping set has full measure. This is highly unexpected, since generically it is believed that for conservative systems hyperbolicity and ellipticity coexist.