# On Herman's Positive Entropy Conjecture

## TL;DR

The paper proves Herman's conjecture by showing that area-preserving surface diffeomorphisms with elliptic fixed points can be perturbed to exhibit positive metric entropy, indicating widespread chaotic behavior.

## Contribution

It demonstrates that the identity map of the disk can be smoothly perturbed to a conservative diffeomorphism with positive entropy, confirming Herman's conjecture and advancing understanding of chaotic dynamics.

## Key findings

- Any area-preserving surface diffeomorphism with an elliptic fixed point can be perturbed to have positive entropy.
- The identity map of the disk can be approximated by diffeomorphisms with positive entropy.
- Existence of a dense set of conservative diffeomorphisms either weakly stable or with chaotic islands.

## Abstract

We show that any area-preserving $C^r$-diffeomorphism of a two-dimensional surface displaying an elliptic fixed point can be $C^r$-perturbed to one exhibiting a chaotic island whose metric entropy is positive, for every $1\le r\le \infty$. This proves a conjecture of Herman stating that the identity map of the disk can be $C^\infty$-perturbed to a conservative diffeomorphism with positive metric entropy. This implies also that the Chirikov standard map for large and small parameter values can be $C^\infty$-approximated by a conservative diffeomorphisms displaying a positive metric entropy (a weak version of Sinai's positive metric entropy conjecture). Finally, this sheds light onto a Herman's question on the density of $C^r$-conservative diffeomorphisms displaying a positive metric entropy: we show the existence of a dense set formed by conservative diffeomorphism which either are weakly stable (so, conjecturally, uniformly hyperbolic) or display a chaotic island of positive metric entropy.