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.[-]

Geodesic flow on ellipsoids is one of the most celebrated classical integrable systems considered by Jacobi in 1837.
Moser revisited this problem in 1978 revealing the link with the modern theory of solitons.
Surprisingly a similar question for hyperboloids did not get much attention, although the dynamics in this case is very different.
I will explain how to use the remarkable results of Moser and Knoerrer on the relations between Jacobi problem and integrable Neumann system on sphere to describe explicitly the geodesic scattering on hyperboloids. It will be shown also that Knoerrer's reparametrisation is closely related to the projectively equivalent metric on a quadric discovered in 1998 by Tabachnikov and, independently, by Matveev and Topalov, giving a new proof of their result. The projectively equivalent metric (in contrast to the usual one) turns out to be regular on the projective closure of hyperboloid, which allows usto extend Knoerrer's map to this closure.
The talk is based on a recent joint work with Lihua Wu.[-]