Self-orbit Equivalences on a Class of Anosov Flows

Abstract Let $M_{k}$ ($k\in{{\mathbb{Z}}}$ and $|k|>4$) be the $3$-manifold obtained by doing $k$ Dehn surgery on the figure-eight knot, and $X_{t}$ be the canonical Anosov flow on $M_{k}$ that is constructed by Goodman. The main result of this article is that if $|k|\gg 0$, then $\mbox{Mod} (M_{k}) \cong{{\mathbb{Z}}}_{2}\oplus{{\mathbb{Z}}}_{2}$ and every element of the mapping class group of $M_{k}$ can be represented by a self orbit equivalence of $X_{t}$.