Report

Teaser, summary, work performed and final results

Periodic Reporting for period 2 - NUHGD (Non Uniform Hyperbolicity in Global Dynamics)

Teaser

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...

Summary

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. 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 (through construction of coding and equilibrium states for instance).

Work performed

We have obtained results towards a global panorama of differentiable dynamical systems, and in particular (1) S. Crovisier, R. Potrie and M. Sambarino have obtained 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 and (2) S. Crovisier and D. Yang have obtained a proof of Palis conjecture about C1 flows far from homoclinic tangencies.

In the setting of surface dynamics, F. Béguin, S. Crovisier and F. Le Roux have contributed to the study of the global dynamics by the description of the maximal isotopies of homeomorphisms (as a complement to Le Calvez theory). S. Crovisier and E. Pujals have introduced the notion of strongly dissipative diffeomorphisms. J. Buzzi, S. Crovisier and O. Sarig have also developed a symbolic description of the dynamics of surface diffeomorphisms with positive entropy, which has allowed to prove Newhouse conjecture about the finiteness of measures of maximal entropy.

We have studied the class of partially hyperbolic dynamics: when the center is one-dimensional, S. Crovisier, D. Yang and J. Zhang have described the existence of physical measures. C. Bonatti and J. Zhang have contributed to the topological classification of these systems. When the center is two-dimensional and conservative, D. Obata and M. Poletti have proved that the fibered Lyapunov exponent is generically positive.

For conservative partially hyperbolic diffeomorphisms, A. Avila, S. Crovisier and A. Wilkinson have proved the Pugh-Shub C1 conjecture about the stable ergodicity of partially hyperbolic diffeomorphisms. D. Obata has built the first example of a C2-stably ergodic partially hyperbolic diffeomorphism whose Oseledts splitting is not dominated.

Final results

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.