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Non Uniform Hyperbolicity in Global Dynamics

Total Cost €


EC-Contrib. €






Project "NUHGD" data sheet

The following table provides information about the project.


Organization address
address: RUE MICHEL ANGE 3
city: PARIS
postcode: 75794

contact info
title: n.a.
name: n.a.
surname: n.a.
function: n.a.
email: n.a.
telephone: n.a.
fax: n.a.

 Coordinator Country France [FR]
 Total cost 1˙229˙255 €
 EC max contribution 1˙229˙255 € (100%)
 Programme 1. H2020-EU.1.1. (EXCELLENT SCIENCE - European Research Council (ERC))
 Code Call ERC-2015-AdG
 Funding Scheme ERC-ADG
 Starting year 2016
 Duration (year-month-day) from 2016-09-01   to  2021-08-31


Take a look of project's partnership.

# participants  country  role  EC contrib. [€] 


 Project objective

An important part of differentiable dynamics has been developed from the uniformly hyperbolic systems. These systems have been introduced by Smale in the 60's in order to address chaotic behavior and are now deeply understood from the qualitative, symbolic and statistic viewpoints. They correspond to the structurally stable dynamics. It appeared that large classes of non-hyperbolic systems also exist. Since the 80's, different notions of relaxed hyperbolicity have been introduced: non-uniformly hyperbolic measures, partial hyperbolicity, ... They allowed to extend the previous approach to other families of systems and to handle new examples of dynamics: the fine description of the dynamics of Hénon maps for instance. The development of local perturbative technics have brought a rebirth for the qualitative description of generic systems. It also opened the door to describe more globally the spaces of differentiable dynamics. For instance, it allowed recent progresses towards the Palis conjecture which characterizes the absence of uniform hyperbolicity by the homoclinic bifurcations — homoclinic tangencies or heterodimensional cycles. We propose in the present project to develop technics for realizing more global perturbations, yielding a breakthrough in the subject. This would settle this conjecture for C1 diffeomorphisms and imply other classification results.

These past years we have understood how qualitative dynamics of generic systems decompose into invariant pieces. We are now ready to describe more precisely the dynamics inside the pieces. We propose to combine these new geometrical ideas to the ergodic theory of non-uniformly hyperbolic systems. This will improve significantly our understanding of general smooth systems (for instance provide existence and finiteness of physical measures and measures of maximal entropy for new classes of systems beyond uniform hyperbolicity).


year authors and title journal last update
List of publications.
2017 Jérôme Buzzi, Sylvain Crovisier, Todd Fisher
Local perturbations of conservative C 1 diffeomorphisms
published pages: 3613-3636, ISSN: 0951-7715, DOI: 10.1088/1361-6544/aa803f
Nonlinearity 30/9 2019-06-13
2018 Sylvain Crovisier, Enrique Pujals
Strongly dissipative surface diffeomorphisms
published pages: 377-400, ISSN: 0010-2571, DOI: 10.4171/cmh/438
Commentarii Mathematici Helvetici 93/2 2019-04-18
2019 Davi Obata
On the stable ergodicity of diffeomorphisms with dominated splitting
published pages: 445-463, ISSN: 0951-7715, DOI: 10.1088/1361-6544/aaea93
Nonlinearity 32/2 2019-04-18
2019 François Béguin, Sylvain Crovisier, Frédéric Le Roux
Projective transformations of rotation sets
published pages: 1237-1241, ISSN: 0951-7715, DOI: 10.1088/1361-6544/aafcde
Nonlinearity 32/4 2019-04-18
2018 Jérôme Buzzi, Sylvain Crovisier, Todd Fisher
The entropy of $C^1$-diffeomorphisms without a dominated splitting
published pages: 6685-6734, ISSN: 0002-9947, DOI: 10.1090/tran/7380
Transactions of the American Mathematical Society 370/9 2019-04-18

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