Self orbit equivalences on the Anosov flows on Dehn surgeries on the figure-eight knot

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 $Mod(M_k)\cong {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$ and every element of the mapping class group of $M_k$ can be represented by a self orbit equivalence of $X_t$.

arXiv comment14 pages, 3 figures. Comparing with version 1, we have corrected a mistake for the choosing of $B_1$ and $B_3$, we have also corrected all related parts. In particular, we have replaced the group ${\mathbb{Z}}_4$ in version 1 by ${\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$