On cyclicity one elliptic islands of the Standard family

TL;DR

This paper investigates the prevalence of cyclicity one elliptic islands in the Standard family of area-preserving maps, showing measure-zero coexistence in most parameters but constructing sets with infinitely many islands at large parameters.

Contribution

It proves that the set of parameters with infinitely many cyclicity one islands has zero Lebesgue measure outside a bounded set and constructs large Hausdorff dimension sets with infinitely many islands.

Findings

Lebesgue measure of parameters with infinite islands is zero outside a bounded set.

Constructed sets with arbitrarily large Hausdorff dimension have infinitely many islands.

Centers of islands accumulate on a hyperbolic set.

Abstract

We study abundance of a special class of elliptic islands (called cyclicity one elliptic islands) for the Standard family of area preserving diffeomorphisms for large parameter values, i.e. far from the KAM regime. Outside a bounded set of parameter values, we prove that the Lebesgue measure of the set of parameter values for which an infinite number of such islands coexist is zero. On the other hand we construct a positive Hausdorff dimension set of arbitrarily large parameter values for which the associated standard map admits infinitely many elliptic islands of cyclicity one, whose centers accumulate on a locally maximal hyperbolic set.