We can conveniently represent post-critically finite topological branched selfcovers of the sphere to itself using maps of graphs. With this representation, there is also a positive characterization of hyperbolic rational maps among these topological branched self-covers, using energies that control elastic 'stretchiness'. In broad terms, a map is rational iff a network of elastic bands gets looser and looser as you pull it back. This complements the older negative characterization of W. Thurston.[-]

In the beginning of the 80s, Masur and Veech proved (independently) that a generic interval exchange transformation is uniquely ergodic. At about the same time, on different sides of the iron curtain, Soviet and French mathematicians got interested in the ergodic properties of measured foliations restricted in some parametric classes. Their motivations and constructions were different but the same fractal objet appeared in both. It is called today the Rauzy gasket.
The motivation from the Soviet side came from the question asked by Sergei Novikov related to the conductivity theory of monocrystals. His question concerns minimal foliations defined by plane sections of surfaces inside the 3-torus. The Rauzy gasket brings an answer to Novikov's question for a surface of genus 3 with central symmetry and 2 double saddles.
Recently, we have obtained a solution of Novikov's problem for any centrally symetric surface of genus 3. This is the most interesting case from the point of view of physical motivations. I will tell some aspects of this story which is based on a series of works, the last and most important one of them is a collaboration with Ivan Dynnikov, Pascal Hubert, Paul Mercat and Alexandra Skripchenko.[-]