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≤∞. 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.