On two-dimensional surface attractors and repellers on 3-manifolds

Abstract

We show that if $f:M^3\to M^3$ is an $A$-diffeomorphism with a surface two-dimensional attractor or repeller $\mathcal{B}$ and $M_{\mathcal{B}}^2$ is a supporting surface for $\mathcal{B}$, then $\mathcal{B}=M_{\mathcal{B}}^2$ and there is $k\ge 1$ such that: 1) $M_{\mathcal{B}}^2$ is a union $M_1^2\cup ...\cup M_k^2$ of disjoint tame surfaces such that every $M_i^2$ is homeomorphic to the 2-torus $T^2$. 2) the restriction of $f^k$ to $M_i^2$ $(i\in \{1,...,k\})$ is conjugate to Anosov automorphism of $T^2$.