Given an Anosov flow \(\phi\) on a 3-manifold \(M\), a periodic orbit \(\gamma\) and an integer \(m\), there is an operation, called Fried surgery, that allows to construct a new 3-manifold \(N\) endowed with an Anosov flow \(\psi\). This operation is a Dehn surgery of slope \(=m\) along the simple closed curve \(\gamma\), but adapted to the pair (Anosov flow, \(3-mfld\)) in such a way that the new manifold is also equipped with an Anosov flow.
Starting from the suspension flow generated by the matrix \(A=[2,1,1,1]\) and making Fried surgeries along its periodic orbits, we can obtain many different Anosov flows, including other suspension flows, some geodesic flows or even flows on hyperbolic 3-manifolds. It is an open problem, however, to determine whether every transitive and orientable Anosov flow can be obtained by doing Fried surgeries on the suspension of \(A=[2,1,1,1]\). The aim of this talk is to show how, by introducing techniques coming from affine geometry on surfaces, we can relate the previous problem with a problem of realizability of quadratic rotations by piecewise affine maps, whose derivatives and breaking points are in a field extension of the rationals by a Perron number.