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Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

5. State of the Art

verfasst von : Gabriel Ponce, Régis Varão

Erschienen in: An Introduction to the Kolmogorov–Bernoulli Equivalence

Verlag: Springer International Publishing

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Abstract

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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Vorheriges Kapitel Hyperbolic Structures and the Kolmogorov–Bernoulli Equivalence
Fußnoten
1
The same argument works if the measure η satisfies the following: if we take a Markov partition \(\{R_i\}_{i=1}^l\) of \(\mathbb T^3\), then we should have η(∂R i) = 0, 1 ≤ i ≤ l. The measure η defined by η = h ∗ m satisfies such property.
 
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Metadaten
Titel
State of the Art
verfasst von
Gabriel Ponce
Régis Varão
Copyright-Jahr
2019
Buch
An Introduction to the Kolmogorov–Bernoulli Equivalence

Print ISBN: 978-3-030-27389-7

Electronic ISBN: 978-3-030-27390-3

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-3-030-27390-3_5