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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. 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 ...

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Documents 11H55 | enregistrements trouvés : 20

Growth and geometry in $SL_2(\mathbb{Z})$ dynamics Veselov, Alexander | CIRM H Nouveau

Lattices and codes:a course partially based on lectures by Friedrich Hirzebruch Ebeling, Wolfgang | Springer Spektrum 2013 Nouveau

the Higher arithmetic:an introduction to the theory of numbers Davenport, H. | Cambridge University Press 2008 Nouveau

La théorie de Markoff et ses développements Perrine, Serge | Tessier & Ashpool 2002 Nouveau

Perfect lattices in euclidean spaces Martinet, Jacques | Springer 2003 Nouveau

Das markovspektrum indefiniter quadratischer formen Martini, Frank | Rheinischen Friedrich-Wilhelms-Universität 1996 Nouveau

Ternäre positiv definite quadratische formen mit gleichen darstellungszahlen Schiemann, Alexander | Rheinischen Friedrich-Wilhelms-Universität 1994 Nouveau

Dynamical systems on homogeneous spaces Starkov, Alexander N. | American Mathematical Society 2000 Nouveau

An introduction to the geometry of numbers Cassels, J. W. S. | Springer 1997 Nouveau

Investigations in number theory Kubota, T. | Academic Press Inc. 1988 Nouveau

I. M. Gelfand seminar Gelfand, Sergei ; Gindikin, Simon | American Mathematical Society 1993 Nouveau

Théorie des nombres Borevitch, Z. I. ; Chafarevitch, I. R. ; Verley, Jean-Luc ; Verley, Myriam | Gauthier-Villars 1967 Nouveau

Quadratic forms Scharlau, Winfried | Queen'S University 1969 Nouveau

Number theory Borevich, Z. I. ; Greenleaf, Newcomb | Academic Press 1967 Nouveau

Lectures on modular forms Gunning, R. C. ; Brumer, Armand | Princeton University Press 1962 Nouveau

An introduction to the geometry of numbers Cassels, J. W. S. | Springer-Verlag 1971 Nouveau

Einführung in die zahlentheorie :introduction to the theory of numbers Dickson, L. E. ; Bodewig, Ewald | Verlag und Druck von B. G. Teubner 1931 Nouveau

Höhere algebra :modern algebraic theories Dickson, L. E. ; Bodewig, Ewald | Verlag und Druck von B. G. Teubner 1929 Nouveau

Rational quadratic forms Cassels, J. W. S. | Academic Press 1978 Nouveau

Une approche géométrique de la réduction des réseaux en petite dimension Vallee, Brigitte | Universite De Caen 1986 Nouveau

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Lattices and codes:
a course partially based on lectures by Friedrich Hirzebruch Ebeling, Wolfgang | Springer Spektrum 2013 Nouveau

the Higher arithmetic:
an introduction to the theory of numbers Davenport, H. | Cambridge University Press 2008 Nouveau

Einführung in die zahlentheorie :
introduction to the theory of numbers Dickson, L. E. ; Bodewig, Ewald | Verlag und Druck von B. G. Teubner 1931 Nouveau

Höhere algebra :
modern algebraic theories Dickson, L. E. ; Bodewig, Ewald | Verlag und Druck von B. G. Teubner 1929 Nouveau