Project

Main goal

The goal of this project is to bring together experts in the varied approaches as well as newcomers including PhD students and postdocs. Consequently, we hope than on one hand we will build long lasting networks of collaboration between the members, and on the other hand we will contribute to the formation of young researchers and students. We plan to organize a workshop in the first year so as to share the expertise and start new collaborations. The second year will be devoted to the ensuing research activities. 

Background and motivation

Lyapunov exponents are key invariants in smooth ergodic theory, often deciding the broad features of the dynamics: both qualitatively (e.g., hyperbolic vs. elliptic, ergodicity, stability) and more quantitatively (e.g., mixing properties including ergodic deviations and spectral gaps). They are usually difficult to control, often failing to be continuous. Various techniques have been developed, some very recently, including: 

• perturbative techniques, e.g., yielding density of hyperbolicity; 

• geometric or analytic methods for quasi-periodic cocycles à la Schrödinger and the consequences for their operator spectrum; 

• invariance principle à la Avila-Viana characterizing their annulation to continuous structures and yielding rigidity, e.g., of the Lyapunov spectrum; 

• regularity from large deviations; continuity “in entropy” related to the so-called strong positive recurrence, a spectral gap property as well as to physical or Sinai-Ruelle-Bowen measures. 

• dichotomy for random diffeomorphism (positive exponents vs. simultaneous linearization). 

These results were often established in special settings (minimal dimension, compact center fibers, cocycles or more general skew-products), raising important questions in more general partially hyperbolic diffeomorphisms with small center dimension and beyond. How these various techniques interact also remains to be explored.

Scientific Objectives

A list of more detailed mathematical questions we plan to address is the following.

• Explore the dichotomy between elliptic and hyperbolic dynamics. For random diffeomorphisms, positive exponents vs simultaneous linearization beyond the spherical case (Dolgopyat-Krikorian, Brown-Rodriguez Hertz). Researchers: Raphael Krikorian, Pablo D. Carrasco, Mauricio Poletti.

• Various rigidity questions in hyperbolic dynamics. Can one extract relevant information about the dynamics from the periodic orbits (the periodic data for Anosov maps, marked length spectrum for Anosov flows) Extend various results in this direction (Gogolev, Tahzibi-Micena, Saghin-Yang for maps, Otal for flows). Researchers: Nicolas Gourmelon, Martin Leguil, Ali Tahzibi, Felipe Riquelme.

• U-Gibbs for partially hyperbolic diffeomorphisms with one-dimensional center. Explore the possibility of extending the recent results (Alvarez-Leguil-Obata-Santiago) in order to understand more general cases, for example when there are positive or zero exponents in the center. Researchers: Martin Leguil, Martin Andersson, Carlos Vázquez, Jiagang Yang. 

• Lyapunov exponents for Anosov maps in higher dimensions. Investigate the continuity of their exponents using perturbation techniques (especially in higher differentiability) and Invariance Principle techniques (Avila-Viana).  Researchers: Jiagang Yang, Karina Marin, Ali Tahzibi, Jerome Buzzi, Radu Saghin, Javier Correa, Mauricio Poletti, Nicolas Gourmelon.

• Measures of maximal entropy for partially hyperbolic diffeomorphisms. There is a satisfactory understanding of the measures of maximal entropy for many classes of partially hyperbolic diffeomorphisms with one dimensional center (Rodriguez Hertz, Tahzibi, Ures, Yang, etc.). Is it possible to go beyond those classes to full generality in this one-dimensional setting (e.g., general discretized Anosov flows)? What about two-dimensional center?  Researchers: Jerome Buzzi, Sylvain Crovisier, Jiagang Yang, Ali Tahzibi, Carlos Vasquez, Radu Saghin, Yuri Lima, Pierre Berger.

• Lyapunov exponents for partially hyperbolic diffeomorphisms with two-dimensional center. Build on the recent progress for diffeomorphisms on surfaces (Buzzi-Crovisier-Sarig) in order to understand the continuity of exponents (with respect to the measure and the map) for some classes of partially hyperbolic diffeomorphisms with two-dimensional center.  Researchers: Jerome Buzzi, Sylvain Crovisier, Karina Marin, Jiagang Yang, Pablo D. Carrasco.

• Relationship between Lyapunov exponents and entropy, especially regarding the SRB measures and on surfaces. Investigate the existence and uniqueness of SRB measures under general conditions and for specific families of endomorphisms on surfaces. Researchers: Martin Andersson, Pablo Carrasco, Radu Saghin, Pierre Berger, David Burguet, Sylvain Crovisier, Jerome Buzzi.

• Lyapunov exponents of quasi-periodic cocycles, and their connection with the spectrum of quasi-periodic Schrödinger operators. Explore the existing techniques to prove positivity of exponents: geometric methods (L.S. Young, Benedicks-Carleson) or methods based on sub-harmonicity and the Avalanche Principle (Bourgain-Goldstein-Schlag, Bourgain-Jitomirskaya, Avila). Researchers: Raphael Krikorian, Martin Leguil

Selected Bibliography

1. J. Buzzi, S. Crovisier, O. Sarig, Continuity properties of Lyapunov exponents for surface diffeomorphisms, arXiv:2103.02400

2. A. Avila, M. Viana. Extremal Lyapunov exponents: an invariance principle and applications. Inventiones mathematicae volume 181, pages 115–178 (2010)

3. R. Saghin, J. Yang, Lyapunov exponents and rigidity of Anosov automorphisms and skew products, Adv. Math. 335 (2019).

4. D Dolgopyat, R Krikorian. Duke Mathematical Journal 136 (3), 475-505, 2007. 5. A. Avila, R. Krikorian. Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles, Annals of Mathematics, Volume 164, 2006, p. 911–940

5. J. Buzzi, S. Crovisier, O. Sarig, Measures of maximal entropy for surface diffeomorphisms. Annals of Mathematics Pages 421-508 from Volume 195 (2022).

6. D. Burguet, Entropy of Physical Measures for C Dynamical Systems, Commun. Math. Phys. 375 (2020), 1201-1222.

7. J. Buzzi, S. Crovisier, O. Sarig, Continuity properties of Lyapunov exponents for surface diffeomorphisms, arXiv:2103.02400