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On Homeomorphisms of Three-Dimensional Manifolds with Pseudo-Anosov Attractors and Repellers

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Abstract

The present paper is devoted to a study of orientation-preserving homeomorphisms on three-dimensional manifolds with a non-wandering set consisting of a finite number of surface attractors and repellers. The main results of the paper relate to a class of homeomorphisms for which the restriction of the map to a connected component of the non-wandering set is topologically conjugate to an orientation-preserving pseudo-Anosov homeomorphism. The ambient \(\Omega\)-conjugacy of a homeomorphism from the class with a locally direct product of a pseudo-Anosov homeomorphism and a rough transformation of the circle is proved. In addition, we prove that the centralizer of a pseudo-Anosov homeomorphisms consists of only pseudo-Anosov and periodic maps.

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Notes

  1. A subspace \(X\) of a topological space \(Y\) is called a cylindrical embedding into \(Y\) of a topological space \(\bar{X}\) if there is a homeomorphism onto the image \(h:\bar{X}\times[-1,1]\to Y\) such that \(X=h(\bar{X}\times\{0\})\).

  2. An invariant set \(B\) of a homeomorphism \(f\) is called an attractor if there is a closed neighborhood \(U\) of the set \(B\) such that \(f(U)\subset{\rm int}U\), \(\underset{j\geqslant 0}{\bigcap}f^{j}(U)=B\). The attractor for the homeomorphism \(f^{-1}\) is called the repeller of the homeomorphism \(f\).

  3. A homeomorphism \(f\) is called periodic if there exists \(m\in\mathbb{N}\) such that \(f^{m}=id\).

  4. Recall that homeomorphisms \(f_{1}\colon X\to X\) and \(f_{2}\colon Y\to Y\) of topological manifolds \(X\) and \(Y\) are called ambiently \(\Omega\)-conjugated if there is a homeomorphism \(h\colon X\to Y\) such that \(h\big{(}NW(f_{1})\big{)}=NW(f_{2})\) and \(hf_{1}|_{NW(f_{1})}=f_{2}h|_{NW(f_{1})}\).

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Funding

The work is supported by the Russian Science Foundation under grant 22-11-00027 except for the results of Section 3 which was supported by the Laboratory of Dynamical Systems and Applications NRU HSE, grant of the Ministry of Science and Higher education of the RF, ag. No. 075-15-2022-1101.

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Authors and Affiliations

  1. International Laboratory of Dynamical Systems and Applications, HSE University, ul. Bolshaya Pecherckaya 25/12, 603155, Nizhny Novgorod, Russia

    Vyacheslav Z. Grines, Olga V. Pochinka & Ekaterina E. Chilina

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  1. Vyacheslav Z. Grines
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  2. Olga V. Pochinka
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  3. Ekaterina E. Chilina
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Correspondence to Olga V. Pochinka or Ekaterina E. Chilina.

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MSC2010

37B99, 37E30

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Grines, V.Z., Pochinka, O.V. & Chilina, E.E. On Homeomorphisms of Three-Dimensional Manifolds with Pseudo-Anosov Attractors and Repellers. Regul. Chaot. Dyn. 29, 156–173 (2024). https://doi.org/10.1134/S1560354724010106

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