Some remarks on the geometry of the Standard Map

TL;DR

This paper introduces a novel geometric framework for the standard map by defining hyperbolic coordinates and foliations, identifying hyperbolic and critical regions, and analyzing their geometric properties.

Contribution

It provides a new geometric perspective on the standard map through hyperbolic coordinates and foliations, highlighting the structure of hyperbolic and critical regions.

Findings

Identification of a uniformly hyperbolic region.

Discovery of a critical region with tangencies between foliations.

Development of a new geometric description of the standard map.

Abstract

We define and compute hyperbolic coordinates and associated foliations which provide a new way to describe the geometry of the standard map. We also identify a uniformly hyperbolic region and a complementary 'critical' region containing a smooth curve of tangencies between certain canonical 'stable' foliations.