On Surface Attractors and Repellers in 3-Manifolds

We show that if $f\colon M^3\to M^3$ is an $A$ diffeomorphism with a surface two-dimensional attractor or repeller $\mathscr B$ with support $M^2_{\mathscr B}$, then $\mathscr B=M^2_{\mathscr B}$ and there exists a $k\ge1$ such that

• 1) $M^2_{\mathscr B}$ is the disjoint union $M^2_1\cup\dots\cup M^2_k$ of tame surfaces such that each surface $M^2_i$ is homeomorphic to the 2-torus $T^2$;
• 2) the restriction of $f^k$ to $M^2_i$, $i\in\{1,\dots,k\}$, is conjugate to an Anosov diffeomorphism of the torus $T^2$.