Continued fractions can be seen in many ways; one of them is giving a directive sequence on a family of dynamical systems. For example, it has been known for a long time that the classical continued fraction is linked to the dynamics of circle rotations, and to the geodesic flow on the modular surface. A more complicated example, the Rauzy-Veech continued fraction and its many variants are linked to the dynamics of families of interval exchange transformations, and to the Teichmüller flow.
We will show that Brun's continued fraction is linked to the dynamics of the translations on the 2-torus, using a coding of these rotations by partitions of the torus with fractal boundaries. The suspension of the natural extension of Brun's continued fraction defines a flow which is a generalisation of the geodesic flow on the modular surface.
This is joint work with Valérie Berthé, Milton Minervino, Wolfgang Steiner and Jörg Thuswaldner.