On Herman's positive entropy conjecture

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 <= infinity. This proves a conjecture of Herman stating that the identity map of the disk can be C-infinity-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-infinity-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. Crown Copyright (C) 2019 Published by Elsevier Inc. All rights reserved.