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On Partially Hyperbolic Diffeomorphisms and Regular Denjoy Type Homeomorphisms

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Abstract

In P. D. McSwiggen’s article, it was proposed Derived from Anosov type construction which leads to a partially hyperbolic diffeomorphism of the 3-torus. The nonwandering set of this diffeomorphism contains a two-dimensional attractor which consists of one-dimensional unstable manifolds of its points. The constructed diffeomorphism admits an invariant one-dimensional orientable foliation such that it contains unstable manifolds of points of the attractor as its leaves. Moreover, this foliation has a global cross section (2-torus) and defines on it a Poincaré map which is a regular Denjoy type homeomorphism. Such homeomorphisms are the most natural generalization of Denjoy homeomorphisms of the circle and play an important role in the description of the dynamics of aforementioned partially hyperbolic diffeomorphisms. In particular, the topological conjugacy of corresponding Poincaré maps provides necessary conditions for the topological conjugacy of the restrictions of such partially hyperbolic diffeomorphisms to their two-dimensional attractors. The nonwandering set of each regular Denjoy type homeomorphism is a Sierpiński set and each such homeomorphism is, by definition, semiconjugate to the minimal translation of the 2-torus. We introduce a complete invariant of topological conjugacy for regular Denjoy type homeomorphisms that is characterized by the minimal translation, which is semiconjugation of the given regular Denjoy type homeomorphism, with a distinguished, no more than countable set of orbits.

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Fig. 1.
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Notes

  1. A basic set \(\Lambda\) is called an attractor (repeller) if it has a closed neighborhood \(U_{\Lambda}\subset M^{3}\) such that \(f(U_{\Lambda})\subset{\rm int}U_{\Lambda}\), \(\bigcap\limits_{k\in\mathbb{N}}f^{k}(U_{\Lambda})=\Lambda\) \((f^{-1}(U_{\Lambda})\subset{\rm int}U_{\Lambda}\), \(\bigcap\limits_{k\in\mathbb{N}}f^{-k}(U_{\Lambda})=\Lambda)\).

  2. Recall that the basic set \(\Lambda\) of an \(A\)-diffeomorphism \(f:M^{3}\rightarrow M^{3}\) is said to be a surface basic set if \(\Lambda\) belongs to an \(f\)-invariant closed surface topologically embedded in the manifold \(M^{3}\).

  3. A nontrivial basic set \(\Lambda\) which is an attractor (repeller) of an \(A\)-diffeomorphism \(f:M^{3}\rightarrow M^{3}\) is said to be expanding (contracting) if the topological dimension of \(\Lambda\) is equal to the dimension of the unstable (stable) manifold of any point of the set \(\Lambda\).

  4. Suppose \(V\) and \(W\) are normed linear spaces, \(A:V\rightarrow W\) is a linear map and \(U\subset V\). Let us define the norm and conorm of \(A\) restricted to \(U\) by: \(||A|U||:=\sup\bigl{\{}\frac{||Av||}{||v||}|v\in U\backslash\{0\}\bigr{\}},||\lfloor A|U\rfloor||:=\inf\bigl{\{}\frac{||Av||}{||v||}|v\in U\backslash\{0\}\bigr{\}}\). According to [16, Definition 2.1], a diffeomorphism \(h\) of a closed three-dimensional manifold \(M^{3}\) is called partially hyperbolic (in a broad sense) if there exists a \(Dh\)-invariant decomposition of the tangent space into two subspaces \(TM^{3}=E^{1}\oplus E^{2}\) and there exist numbers \(0<\lambda<\mu,c>0\) such that for all points \(x\in M^{3}\) the following holds: \(||Dh^{n}|_{E^{1}_{x}}||\leqslant c\lambda^{n}\) and \(c^{-1}\mu^{n}\leqslant||\lfloor Dh^{n}|_{E^{2}_{x}}\rfloor||\), where \(n\in\mathbb{N}\).

  5. Here by the Cantor set we mean a perfect nowhere dense set on the circle.

  6. Here by \(p^{-1}(x)\) one means the complete preimage of the point \(x\).

  7. The map \(\varphi:X\rightarrow Y\), where \(X,Y\) are topological spaces, is said to be an embedding if \(\varphi:X\rightarrow\varphi(X)\subset Y\) is a homeomorphism, where \(\varphi(X)\) carries the induced topology inherited from \(Y\). Here by closed embedded disk we mean an image of a closed disk \(D=\{(x_{1},x_{2})\in\mathbb{R}^{2}|x_{1}^{2}+x_{2}^{2}\leqslant 1\}\) with respect to the embedding \(\tau:D\rightarrow\mathbb{T}^{2}\).

  8. Let \(X\) be a metric space with the metric \(\rho\) and \(A\) be a subset of this space. The diameter of the set \(A\) is the following value: \(\mathrm{diam}A=\sup\{\rho(x,y):x,y\in A\}\).

  9. In the one-dimensional case, in accordance with [9, p. 348], it is possible to construct a Denjoy homeomorphism of the circle with a characteristic set consisting of a countable number of orbits.

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Funding

The final version of the article was obtained with the financial support from the RSF grant (project 21-11-00010) using materials previously obtained with the financial support from the RSF grant (project 17-11-01041). In addition, the proof of Theorem 2 was obtained with the financial support from 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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  1. HSE University, ul. Bolshaya Pecherskaya 25/12, 603155, Nizhny Novgorod, Russia

    Vyacheslav Z. Grines & Dmitrii I. Mints

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  1. Vyacheslav Z. Grines
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Correspondence to Vyacheslav Z. Grines or Dmitrii I. Mints.

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37E30, 37D30

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Grines, V.Z., Mints, D.I. On Partially Hyperbolic Diffeomorphisms and Regular Denjoy Type Homeomorphisms. Regul. Chaot. Dyn. 28, 295–308 (2023). https://doi.org/10.1134/S1560354723030036

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