In this talk we introduce several properties of (special) diffeomorphisms of surfaces. More precisely, we discuss a result of R. Brown in which proves non-ergodicity of a single pseudo-Anosov homeomorphism on some relative character varieties. The SU(2)-character variety of a genus g surface with b punctures is the space of representations from the fundamental group of the surface to the special unitary group SU(2) up to conjugacy by elements of SU(2). In fact, the mapping class group of the surface act on the character veriety by pre-composition. After having introduced the different concepts involved, we discuss this result and give some generalizations on the character variety of the punctured torus and the four-punctured sphere and explain how to use these results to construct a family of pseudo-Anosov homeomorphisms of the twice-punctured torus with elliptic fixed points.