Spectrum of SL(2,R)-characters: the once-punctured torus case

Consider a topological surface $\mathrm{\Sigma }$. We introduce the spectrum of a representation from the fundamental group of $\mathrm{\Sigma }$ to SL(2,R), which is a subset of projective measured lamination on the surface, which captures the directions along which the representation fails to be Fuchsian, and which characterizes the action of the mapping class group on this representation. In the case of the once-punctured torus, we show that the spectrum of a generic representation is a Cantor set, and that it completely describes the dynamics of the familly of locally constant cocycles above interval exchange transformations associated to the representation.