Spectral properties of linear involutions

Linear involutions form a natural class of one-dimensional dynamical systems arising from non-orientable foliations on orientable surfaces, and can be viewed as a generalization of interval exchange transformations. In this talk, I will introduce linear involutions and discuss their basic dynamical properties, highlighting similarities and differences with the classical theory of interval exchanges. I will then focus on their spectral behavior, in particular on weak mixing phenomena. I will present results establishing weak mixing for typical linear involutions and explain the main ideas and mechanisms underlying the proof.