On Herman's Positive Entropy Conjecture

2017/04/08 by Berger, Pierre, Turaev, Dimitry · 2 citations
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1704.02473

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 diffeomorphism which either are weakly stable (so, conjecturally, uniformly hyperbolic) or display a chaotic island of positive metric entropy.

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