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

Viatcheslav Grines·Vladislav Medvedev·Evgeny Zhuzhoma
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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^2_ \mathcal B$ is a supporting surface for $ \mathcal B$, then $\mathcal B = M^2_{\mathcal B}$ and there is $k\geq 1$ such that: 1) $M^2_{\mathcal B}$ is a union $M^2_1\cup...\cup M^2_k$ of disjoint tame surfaces such that every $M^2_i$ is homeomorphic to the 2-torus $T^2$. 2) the restriction of $f^k$ to $M^2_i$ $(i\in\{1,...,k\})$ is conjugate to Anosov automorphism of $T^2$.

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Viatcheslav Grines, Vladislav Medvedev, Evgeny Zhuzhoma. 2004-12-16. On two-dimensional surface attractors and repellers on 3-manifolds. https://doi.org/10.1007/s11006-005-0181-1

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