Let Mk (k 2 Z and jkj > 4) be the 3-manifold obtained by doing k Dehn surgery on the gure-eight knot, and Xt be the canonical Anosov ow on Mk that is constructed by Goodman. In this talk, I will explain a brief proof of the following 40 result: if jkj ≫ 0, then Mod(Mk) = Z4 and every element of the mapping class group of Mk can be represented by a self orbit equivalence of Xt. This answered a question asked by Barthelme and Mann.