## 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≤r≤∞. This proves a conjecture of Herman stating that the identity map of the disk can be C ^{∞} -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 ^{∞} -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 diffeomorphisms which either are weakly stable (so, conjecturally, uniformly hyperbolic) or display a chaotic island of positive metric entropy.

Original language | English |
---|---|

Pages (from-to) | 1234-1288 |

Number of pages | 55 |

Journal | Advances in Mathematics |

Volume | 349 |

DOIs | |

State | Published - 20 Jun 2019 |

Externally published | Yes |

## Keywords

- Homoclinic biffurcation
- Lyapunov exponents
- Metric entropy
- Symplectic dynamics
- Universal map

## ASJC Scopus subject areas

- General Mathematics