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On the Number of the Classes of Topological Conjugacy of Pixton Diffeomorphisms

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Abstract

For a wide class of dynamical systems known as Pixton diffeomorphisms the topological conjugacy class is completely defined by the Hopf knot equivalence class, i.e. the knot whose equivalence class under homotopy of the loops is a generator of the fundamental group \(\pi _1(S^2\times S^1)\). Moreover, any Hopf knot can be realized by a Pixton diffeomorphism. Nevertheless, the number of the classes of topological conjugacy of these diffeomorphisms is still unknown. This problem can be reduced to finding topological invariants of Hopf knots. In the present paper we describe a first order invariant for these knots. This result allows one to model countable families of pairwise non-equivalent Hopf knots and, therefore, infinite set of topologically non-conjugate Pixton diffeomorphisms.

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Notes

  1. C. Bonatti and V.Z. Grines knew that there exists a countable set of pairwise non-equivalent Hopf knots. For the first time this fact was mentioned by I.V. Itenberg (who discussed the subject with O.Ya. Viro and learned from him about Mazur knot) to V.Z. Grines and later it was confirmed by V.A. Vasilyev to E.V. Zhuzhoma during their meeting in Rennes.

  2. The idea of the proof follows from the results of Section 4.1 [3] formulated in terms of dynamical systems.

  3. Notice that in [2] there is an invariant of the first order for the contractible knot \(L \subset S^2 \times S^1\) (this case is included in the more general one). But Theorem 1 does not follow from these results.

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Acknowledgements

This work was supported by the Russian Science Foundation (project 21-11-00010). The first author thanks S.A. Melikhov and V.V. Chernov for fruitful discussions.

Funding

This work was supported by the Russian Science Foundation (project 21-11-00010).

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Authors and Affiliations

  1. HSE Tikhonov Moscow Institute of Electronics and Mathematics (MIEM HSE), Moscow, Russia

    P. M. Akhmet’ev

  2. Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation Russian Academy of Sciences (IZMIRAN), Moscow, Russia

    P. M. Akhmet’ev

  3. Laboratory of Algorithms and Technologies for Network Analysis, National Research University Higher School of Economics, 136 Rodionova Street, Niznhy Novgorod, Russia

    T. V. Medvedev

  4. National Research University Higher School of Economics, 25/12 Bolshaya Pecherckaya Street, Niznhy Novgorod, Russia

    O. V. Pochinka

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  1. P. M. Akhmet’ev
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  2. T. V. Medvedev
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  3. O. V. Pochinka
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Correspondence to T. V. Medvedev.

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Akhmet’ev, P.M., Medvedev, T.V. & Pochinka, O.V. On the Number of the Classes of Topological Conjugacy of Pixton Diffeomorphisms. Qual. Theory Dyn. Syst. 20, 76 (2021). https://doi.org/10.1007/s12346-021-00518-1

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