On Herman's positive entropy conjecture

Pierre Berger, Dmitry Turaev

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

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 languageEnglish
Pages (from-to)1234-1288
Number of pages55
JournalAdvances in Mathematics
Volume349
DOIs
StatePublished - 20 Jun 2019
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • General Mathematics

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