Non Uniform Hyperbolicity in Global Dynamics

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 characterize the absence of uniform hyperbolicity by the homoclinic bifurcations — homoclinic tangencies or heterodimensional cycles. The present project develops technics for realizing more global perturbations, yielding a breakthrough in the subject. This would allow to 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 improves significantly our understanding of general smooth systems (through construction of coding and equilibrium states for instance).

Among the results obtained, major progress have occurred on:
- dissipative surface dynamics and renormalization of entropy zero Hénon maps,
- measures maximizing the entropy for surface diffeomorphisms,
- coding and quantitative ergodic properties of non-uniformly hyperbolic systems,
- stable ergodicity of partially hyperbolic systems,
- attractors and empirical measures of differentiable dynamics far from homoclinic tangencies.

We have obtained results towards a global panorama of differentiable dynamical systems, and in particular
- a proof of Palis conjecture about C1 flows far from homoclinic tangencies,
- a partial answer to Bonatti’s conjecture about the finiteness of the number of chain-recurrence classes for C1-generic diffeomorphisms far from homoclinic tangencies,
- a characterization of the class of discretized Anosov flows,
- a study of multi-singular hyperbolic flows in higher dimension.

In the setting of surface dynamics, we have developed tools allowing to address critical behaviors:
- introduction of the class of mildly dissipative dynamics in order to renormalize Hénon maps with zero entropy,
- interaction between homoclinic tangencies and heterodimensional cycles, producing germ-typical Newhouse phenomenon.

The fine structure and properties of the non-uniformly hyperbolic locus have been studied, through codings and Yomdin’s technique. This has lead to results on:
- finiteness of homoclinic classes and measures maximizing the entropy,
- continuity of Lyapunov exponents, and introduction of a new class of non-uniform hyperbolicity,
- quantitative statistical properties of equilibrium measures.

Results on physical and geometrical measures have been obtained:
- existence and rigidity of u-Gibbs states in the partially hyperbolic setting,
- existence of SRB measures for surface diffeomorphisms (Viana’s conjecture),
- invariance principle and criterion for non-vanishing Lyapunov exponents of partially hyperbolic systems with non-compact center bundle,
- stable ergodicity of generic conservative partially-hyperbolic systems.

Some interactions with other dynamical settings have also been explored (geodesic flows, billiards, interval exchange transformations).

This has lead to about 50 research papers and the organization of 8 meetings.

The collaboration Crovisier-Potrie-Sambarino has developed the first global C1-perturbations for the dynamics of partially hyperbolic systems. We expect to deepen this technique in order to address the Palis conjecture.
The work by Buzzi-Crovisier-Sarig has introduced the measured homoclinic classes for non-uniformly hyperbolic systems. We expect to obtain consequences about equilibrium states. S. Crovisier and D. Yang have built fibered models for analyzing the local dynamics of vector fields. Buzzi-Crovisier-Lima propose to implement this tool for studying the symbolic dynamics of non-uniformly hyperbolic flows in dimension 3.

Periodic Reporting for period 5 - NUHGD (Non Uniform Hyperbolicity in Global Dynamics)

Diese Seite teilen

Herunterladen