On stochastic sea of the standard map

TL;DR

This paper demonstrates that for many parameters, the stochastic sea of the standard map has full Hausdorff dimension, revealing complex invariant sets and the accumulation of elliptic islands.

Contribution

It proves that the stochastic sea of the standard map has full Hausdorff dimension for large generic parameters, linking homoclinic tangencies to invariant set complexity.

Findings

The invariant set  has full Hausdorff dimension.

The invariant set  is a topological limit of hyperbolic sets.

The stochastic sea is densely populated by elliptic islands.

Abstract

Consider a generic one-parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism. We show that for many parameters (residual subset in an open set approaching the critical value) the corresponding diffeomorphism has a transitive invariant set $\Omega$ of full Hausdorff dimension. The set $\Omega$ is a topological limit of hyperbolic sets and is accumulated by elliptic islands. As an application we prove that stochastic sea of the standard map has full Hausdorff dimension for sufficiently large topologically generic parameters.