Hyperbolic dynamical systems

 

In Dynamic Systems, my culture, my geometric aesthetics and the kindness of Rémi Langevin first naturally led me to the study of hyperbolic systems.

 

What is a hyperbolic system?

The simplest chaotic dynamical system is the multiplication by 10 (or by any integer ≥ 2) on the circle. One easily checks that there are periodic points of arbitrary large periods arbitrarily close to any point and that there are points whose orbit is dense. By definition any error on the initial data is multiplied by 10 at each iteration: that is the sensitivity to initial conditions. A somewhat paradoxical theorem by Shub [Sh1969] says that every perturbation of this system keeps it unchanged: by writing the new system in a reparametrization of the circle, one recovers exactly the multiplication by 10: one says that the dynamics is structurally stable. It is a toy model but, indeed, every chaotic dynamics admits some part which behaves as the multiplication by 10.

The multiplication by 10 on the circle is not invertible: one cannot divide by 10 on the circle.  If one wants an invertible system (where the orbits are defined in the past and in the future) the expanding directions need to be balanced by contracting directions. That is precisely what happens when one considers the action on the torus T²=R²/Z² of the linear map of R² x→2x+y, y→x+y which has one eigenvalue larger than 1 and another one smaller than 1 (corresponding to an expanding and a contracting direction).

This is still a toy model which is the starting point of Smale’s hyperbolic theory in the ‘60s.  The main achievement of this theory is that hyperbolic systems (those for which the tangent space at the recurrent orbits is the sum of the contracting and expanding directions) are precisely the perfectly chaotic systems that are structurally stable (the proof of this theorem took from the ‘60s to the end ‘90s, (Smale [Sm1967], Robbin [Ro1971], Robinson [Ro1976], Mañé [Ma1988], Hayashi [Ha1998]).

These are not non-sense mathematical objects: Poincaré and Hadamard (around 1900) studied indeed an emblematic example as a model for the three-body problem of celestial mechanics: the geodesic flow of hyperbolic surfaces. Hadamard already expressed its chaotic behavior with a modern formulation: « tout changement, si minime qu’il soit, apporté à la direction initiale d’une géodésique (…) suffit pour amener une variation absolument quelconque dans l’allure finale de la courbe »

“Any change, no matter how small, made to the initial direction of a geodesic (...) is sufficient to bring about an entirely different final trajectory of the curve.” ([Ha1898]).

Anosov and Smale have defined the notion of hyperbolic system by extracting the fundamental principle behind the geodesic flow of hyperbolic surfaces. A system is said to be Anosov if the contraction/expansion structure is filling the whole manifold so that the dynamics in a neighborhood of any orbit behaves as in Figure 1.

 

 

The classification problem

With Langevin we sought to give a topological classification of hyperbolic systems.

Any classification problem is broken down into three steps:

- Give a finished presentation of the objects studied.

- Give a criterion or failing that an algorithm making it possible to decide which finished presentations correspond to the same object.

 - Say which abstract finite presentations actually correspond to an object.

 

Smale diffeomorphisms of compact surfaces

With R.Langevin, I gave a finite presentation of “hyperbolic” diffeomorphisms of compact surfaces, i.e. verifying axiom A and strong transversality (= resolution of the first classification problem). This result encompasses a whole family of previous partial results from a Russian school around Aranson and Grines.

With E. Jeandenans I characterized the finite presentations corresponding to diffeomorphisms of surfaces (=resolution of the third problem).

These results are the subject of Asterisque 250 (reference 32 below)

 

François Beguin completed the classification of hyperbolic diffeomorphisms of surfaces by providing a finite algorithm for deciding when two finite presentations correspond to the same diffeomorphism (resolution of the second problem).

 

 

Smale flows of manifolds of dimension 3

With Fr. Beguin I continued this research by studying the hyperbolic flows of 3-dimensional manifolds. This research combines the difficulties of surface diffeomorphisms with specific difficulties due to the topology of 3-dimensional manifolds.

We have shown how to give a finite presentation of structurally stable vector fields of 3-dimensional manifolds (without non-periodic attractors/repulsors). We have shown that all finite presentations correspond to flows (= first and third classification problem) (see 27 and 31 in the list below).

 

The second problem ("deciding when two finite presentations correspond to the same flow)" seems to be much more complex: in the particular case of Anosov flows, many researchers have tackled it, without concluding, until now (see the work of T. Barbot and S. Fenley).  Similarly, Christy has announced a comprehensive classification of hyperbolic attractors for over 30 years, but his work has never been written.

 This question is now my main personal challenge, because it associates dynamics with the topology of 3-dimensional manifolds. See [1,2,3,4,6,13,14,27,31,34] in the list below below, for my work on the Anosov flows.

 

Morse Smale diffeomorphisms (= hyperbolic but not chaotic) of 3-manifolds

The complexity of the problem already in dimension 2 and for the flows in dimension 3 seems to mean that a classification of hyperbolic diffeomorphisms in dimension 3 is for the moment out of reach... Moreover, hyperbolic dynamics in dimension 3 not being not dense, one might wonder what the stakes of such an undertaking would be...

However, I was surprised to see at what elementary level (from the dynamic point of view) the topological difficulties arrived: thus with V. Grines we showed in 2000 that the classification of Morse Smale diffeomorphisms whose non-wandering set is reduced to 4 fixed points (two repulsors, a saddle and an attractor) is equivalent to the classification of nodes in S2x S1, freely homotopic to {0}xS1 (see reference 30 below).

Then, with Grines and Pochinka and other collaborators like Laudenbach, we gradually considered more and more general cases in a series of 15 articles until we obtained a complete and canonical conjugation invariant for all Morse-Smale diffeomorphisms ( hyperbolic but not chaotic)  on manifolds of dimension 3.

 

List of my publications in hyperbolic dynamics

  1. 1.Barthelmé, Thomas; Bonatti, Christian; Mann, Kathryn Promoting Prelaminations.  Preprint ArXiv:2406.18917  (2024) 

  2. 2.Bonatti, Christian; Iakovoglou, Ioannis Anosov flows on 3-manifolds: the surgeries and the foliations.  Ergodic Theory Dyn. Syst. 43, No. 4, 1129-1188 (2023). 

  3. 3.Bonatti, Christian Action on the circle at infinity of foliations of ℝ2    arXiv:2301.04530 (2023) Enseign. Math. (2024), published online first   DOI 10.4171/LEM/1086 

  4. 4.Asaoka, Masayuki; Bonatti, Christian; Marty, Théo Oriented Birkhoff sections of Anosov flows. Preprint, arXiv:2212.06483   (2022). 

  5. 5.Bonatti, Christian; Pinsky, Tali Lorenz attractors and the modular surface.  Nonlinearity 34, No. 6, 4315-4331 (2021). 

  6. 6.Barthelmé, Thomas; Bonatti, Christian; Gogolev, Andrey; Rodriguez Hertz, Federico Anomalous Anosov flows revisited.  Proc. Lond. Math. Soc. (3) 122, No. 1, 93-117 (2021). 

  7. 7.Bonatti, Christian; Bowden, Jonathan; Potrie, Rafael Some remarks on projective Anosov flows in hyperbolic 3-manifolds.  Wood, David R. (ed.) et al., 2018 MATRIX annals. Cham: Springer. MATRIX Book Ser. 3, 359-369 (2020). 

  8. 8.Bonatti, Christian; Minkov, Stanislav; Okunev, Alexey; Shilin, Ivan Anosov diffeomorphism with a horseshoe that attracts almost any point.  Discrete Contin. Dyn. Syst. 40, No. 1, 441-465 (2020). 

  9. 9.Bonatti, C.; Grines, V.; Pochinka, O. Topological classification of Morse-Smale diffeomorphisms on 3-manifolds.  Duke Math. J. 168, No. 13, 2507-2558 (2019). 

  10. 10.Bonatti, Ch.; Grines, V.; Laudenbach, F.; Pochinka, O.Topological classification of Morse-Smale diffeomorphisms without heteroclinic curves on 3-manifolds.  Ergodic Theory Dyn. Syst. 39, No. 9, 2403-2432 (2019). 

  11. 11.Bonatti, C.; Minkov, S. S.; Okunev, A. V.; Shilin, I. S. A C1Anosov diffeomorphism with a horseshoe that attracts almost any point. Funct. Anal. Appl. 51, No. 2, 144-147 (2017); translation from Funkts. Anal. Prilozh. 51, No. 2, 83-86 (2017). 

  12. 12.Bonatti, Ch.; Grines, V. Z.; Pochinka, Olga V. Realization of Morse-Smale diffeomorphisms on 3-manifolds. Proc. Steklov Inst. Math. 297, 35-49 (2017); translation from Tr. Mat. Inst. Steklova 297, 46-61 (2017). 

  13. 13.Béguin, François; Bonatti, Christian; Yu, Bin Building Anosov flows on 3-manifolds.  Geom. Topol. 21, No. 3, 1837-1930 (2017). 

  14. 14.Beguin, F.; Bonatti, C.; Yu, B. A spectral-like decomposition for transitive Anosov flows in dimension three.  Math. Z. 282, No. 3-4, 889-912 (2016). 

  15. 15.Bonatti, Christian; Guelman, Nancy Axiom A diffeomorphisms derived from Anosov flows.  J. Mod. Dyn. 4, No. 1, 1-63 (2010). 

  16. 16.Bonatti, Christian; Guelman, Nancy Transitive Anosov flows and Axiom-A diffeomorphisms. Ergodic Theory Dyn. Syst. 29, No. 3, 817-848 (2009). 

  17. 17.Bonatti, C.; Paoluzzi, L. 3-manifolds which are orbit spaces of diffeomorphisms. Topology 47, No. 2, 71-100 (2008). 

  18. 18.Bonatti, C.; Grines, V. Z.; Medvedev, V. S.; Pochinka, O. V. Bifurcations of Morse-Smale diffeomorphisms with wildly embedded separatrices.  Proc. Steklov Inst. Math. 256, 47-61 (2007); translation from Tr. Mat. Inst. Steklova 256, 54-69 (2007). 

  19. 19.Bonatti, Christian; Grines, Viacheslav; Pochinka, Olga Classification of Morse-Smale diffeomorphisms with the chain of saddles on 3-manifolds.  Walczak, Paweł (ed.) et al., Foliations 2005. Proceedings of the international conference, University of Łodź, Łodź, Poland, June 13–24, 2005. Hackensack, NJ: World Scientific (ISBN 981-270-074-9). 121-147 (2006).  

  20. 20.Bonatti, C.; Grines, V. Z.; Pochinka, O. V. Classification of Morse-Smale diffeomorphisms with a finite set of heteroclinic orbits on 3-manifolds.  Proc. Steklov Inst. Math. 250, 1-46 (2005); translation from Tr. Mat. Inst. Steklova 250, 5-53 (2005). 

  21. 21.Bonatti, C.; Grines, V.; Pochinka, O. Classification of the simplest non-gradient-like diffeomorphisms on 3-manifolds. J. Math. Sci., New York 126, No. 4, 1267-1296 (2005). 

  22. 22.Bonatti, Ch.; Grines, V. Z.; Pochina, O. V. Classification of Morse-Smale diffeomorphisms with finite sets of heteroclinic orbits on 3-manifolds.  Dokl. Math. 69, No. 3, 385-387 (2004); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 396, No. 4, 439-442 (2004). 

  23. 23.Bonatti, C.; Grines, V.; Medvedev, V.; Pécou, E. Topological classification of gradient-like diffeomorphisms on 3-manifolds.  Topology 43, No. 2, 369-391 (2004). 

  24. 24.Bonatti, C.; Grines, V. Z.; Medvedev, V. S.; Pécou, E. On Morse-Smale diffeomorphisms without heteroclinic intersections on three-manifolds. Proc. Steklov Inst. Math. 236, 58-69 (2002); translation from Tr. Mat. Inst. Steklova 236, 66-78 (2002). 

  25. 25.Bonatti, Christian; Grines, Viatcheslav; Pécou, Elisabeth Two-dimensional links and diffeomorphisms on 3-manifolds.  Ergodic Theory Dyn. Syst. 22, No. 3, 687-710 (2002). 

  26. 26.Bonatti, C.; Grines, V.; Medvedev, V.; Pécou, E. Three-manifolds admitting Morse-Smale diffeomorphisms without heteroclinic curves.  Topology Appl. 117, No. 3, 335-344 (2002). 

  27. 27.Béguin, F.; Bonatti, C. Flots de Smale en dimension 3: Présentations finies de voisinages invariants d’ensembles selles. Topology 41, No. 1, 119-162 (2002). 

  28. 28.Bonatti, C.; Grines, V.; Langevin, R. Dynamical systems in dimension 2 and 3: conjugacy invariants and classification.  Comput. Appl. Math. 20, No. 1-2, 11-50 (2001). 

  29. 29.Bonatti, C.; Grines, V. Z.; Medvedev, V. S.; Pecou, E. On the topological classification of gradient-like diffeomorphisms without heteroclinic curves on three-dimensional manifolds. Dokl. Math. 63, No. 2, 161-164 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 377, No. 2, 151-155 (2001). 

  30. 30.Bonatti, C.; Grines, V. Knots as topological invariants for gradient-like diffeomorphisms of the sphere S3  J. Dyn. Control Syst. 6, No. 4, 579-602 (2000). 

  31. 31.Béguin, F.; Bonatti, C.; Vietez, J. L. Construction de flots de Smale en dimension 3. Ann. Fac. Sci. Toulouse, VI. Sér., Math. 8, No. 3, 369-410 (1999). 

  32. 32.Bonatti, C.; Langevin, R. [Jeandenans, E.] Difféomorphismes de Smale des surfaces. Avec la collaboration d’Emmanuelle Jeandenans.  Astérisque. 250. , viii, 235 p. (1998). 

  33. 33.Bonatti, Christian; Díaz, Lorenzo Justiniano; Viana, Marcelo Discontinuity of the Hausdorff dimension of hyperbolic sets.  C. R. Acad. Sci., Paris, Sér. I 320, No. 6, 713-718 (1995). 

  34. 34.Bonatti, Christian; Langevin, Remi Un exemple de flot d’Anosov transitif transverse à un tore et non conjugué à une suspension. Ergodic Theory Dyn. Syst. 14, No. 4, 633-643 (1994). 

 

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