No elliptic islands for the universal area-preserving map

TL;DR

This paper proves, using computer-aided methods, that the universal area-preserving map has no elliptic islands with period less than 20 and that less than 1.5% of its domain contains elliptic islands, suggesting predominantly hyperbolic behavior.

Contribution

It provides the first computer-aided proof that no small-period elliptic islands exist in the universal area-preserving map and estimates the measure of elliptic regions.

Findings

No elliptic islands with period less than 20 found.

Less than 1.5% of the domain contains elliptic islands.

At least 98.5% of the domain's measure escapes to infinity.

Abstract

A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} to prove the existence of a \textit{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 \cite{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.