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H 1 Growth and geometry in $SL_2(\mathbb{Z})$ dynamics

Auteurs : Veselov, Alexander (Auteur de la Conférence)
CIRM (Editeur )

Résumé : Usual discrete dynamics can be considered as the action of the group of integers. What happens if we replace $\mathbb{Z}$ by $SL_2(\mathbb{Z})$?
There is a classical example of such dynamics goes back to remarkable work by Andrei A. Markov (1880), who described the solutions of the Diophantine equation $x^2 + y^2 + z^2 = 3xyz$ (known now as Markov triples) as an orbit of $SL_2(\mathbb{Z})$. These triples surprisingly appeared in many areas of mathematics: initially in arithmetic, but more recently in hyperbolic and algebraic geometry, the theory of Teichmüller spaces, Frobenius manifolds and Painlevé equations.
Another example of such dynamics appears in the description of the values of a binary quadratic form $Q(x,y) = ax^2+bxy+cy^2$ with integer coefficients, the problem going back to Gauss. About 20 years ago John H. Conway proposed a ”topographic” approach to this problem, using the planar trivalent tree, which can be considered as a discrete version of the hyperbolic plane.
The same approach can be applied to general $SL_2(\mathbb{Z})$ dynamics, and in particular to Markov dynamics as well. The growth of the corresponding numbers depends on the paths on such tree, which can be labelled by the points of real projective line.
I will discuss some results about the corresponding Lyapunov exponents found jointly with K. Spalding and A. Sorrentino, using the known links with the hyperbolic geometry.

Keywords : Markov numbers; Lyapunov exponents; Farey tree; Conway river; Markov spectrum

Codes MSC :
11H55 - Quadratic forms (reduction theory, extreme forms, etc.)
11J06 - Markov and Lagrange spectra and generalizations
34D08 - Characteristic and Lyapunov exponents

 Informations sur la Vidéo Langue : Anglais Date de publication : 11/10/2018 Date de captation : 04/10/2018 Sous collection : Research talks Format : MP4 arXiv category : Dynamical Systems ; Number Theory Domaine : Number Theory ; Dynamical Systems & ODE Durée : 00:31:42 Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2018-10-04_Veselov.mp4 Informations sur la rencontre Nom de la rencontre : 6th International conference on uniform distribution theory - UDT2018 / 6e Colloque international sur la théorie de la répartition uniforme - UDT2018Organisateurs de la rencontre : Karpenkov, Oleg ; Nair, Radhakrishnan ; Verger-Gaugry, Jean-LouisDates : 01/10/2018 - 05/10/2018 Année de la rencontre : 2018 URL Congrès : https://conferences.cirm-math.fr/1860.htmlCitation Data DOI : 10.24350/CIRM.V.19455003 Cite this video as: Veselov, Alexander (2018). Growth and geometry in $SL_2(\mathbb{Z})$ dynamics. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19455003 URI : http://dx.doi.org/10.24350/CIRM.V.19455003

### Voir aussi

Bibliographie

1. Sorrentino, A., & Veselov, A.P. (2017). Markov Numbers, Mather's β function and stable norm. - https://arxiv.org/abs/1707.03901

2. Spalding, K., & Veselov, A.P. (2018). Growth of values of binary quadratic forms and Conway rivers. Bulletin of the London Mathematical Society, 50(3), 513-528 - https://doi.org/10.1112/blms.12156

3. Spalding, K., & Veselov, A.P. (2018). Veselov Conway river and Arnold sail. - https://arxiv.org/abs/1801.10072

4. Spalding, K., & Veselov, A.P. (2017). Tropical Markov dynamics and Cayley cubic. - https://arxiv.org/abs/1707.01760

5. Spalding, K., & Veselov, A.P. (2017). Lyapunov spectrum of Markov and Euclid trees. Nonlinearity, 30(12), 4428-4453 - https://doi.org/10.1088/1361-6544/aa88ff

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