A hyperbolic 2x2 matrix of determinant one whose entries lie in a field is called "special" if its eigenvalues lie in the field. (Generically, such eigenvalues lie in a quadratic extension.) We exhibit examples of special matrices that correspond to "pseudo Anosov" diffeomorphisms on "flat" surfaces. These diffeomorphisms are also special: they define "stable flow directions" whose dynamics are special. In particular, the "Sah-Arnoux-Fathi invariant" vanishes for these flows. I will sketch the motivating dynamics, and focus on number theoretic considerations. The results are joint work with K. Calta.