# Lyapunov exponents for random perturbations of some area-preserving maps including the standard map

## TL;DR

This paper demonstrates that small random perturbations of certain 2D area-preserving maps, including the Standard map, allow for easier estimation of Lyapunov exponents, revealing the maps' underlying geometric properties.

## Contribution

It introduces a method to estimate Lyapunov exponents for non-uniformly hyperbolic area-preserving maps using small random perturbations.

## Key findings

- Random perturbations facilitate Lyapunov exponent estimation.
- Lyapunov exponents reflect the geometry of the original maps.
- Applicable to maps with strong hyperbolicity on large regions.

## Abstract

We consider a large class of 2D area-preserving diffeomorphisms that are not uniformly hyperbolic but have strong hyperbolicity properties on large regions of their phase spaces. A prime example is the Standard map. Lower bounds for Lyapunov exponents of such systems are very hard to estimate, due to the potential switching of "stable" and "unstable" directions. This paper shows that with the addition of (very) small random perturbations, one obtains with relative ease Lyapunov exponents reflecting the geometry of the deterministic maps.