State of the Art

  • Gabriel Ponce
  • Régis Varão
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)


This chapter is devoted to present some other results concerning the equivalence of the Kolmogorov and the Bernoulli property for systems which preserve a smooth measure and admit some level of hyperbolicity. We define the class of non-uniformly hyperbolic diffeomorphisms (resp. flows), the class of smooth maps (resp. flows) with singularities, and the class of partially hyperbolic diffeomorphisms derived from Anosov, and present the state of art of the problem inside each of this classes. In each case we briefly comment on the similarities with the Anosov case as well as the central difficulties that appear along the arguments. The class derived from Anosov diffeomorphisms is the one for which the results differ the most from the results for Anosov diffeomorphisms, therefore we go deeper in this particular case and prove the key results which allow us to overcome the absence of complete hyperbolicity along the center direction.


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© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Gabriel Ponce
    • 1
  • Régis Varão
    • 1
  1. 1.IMECCUniversity of Campinas - UNICAMPCampinasBrazil

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