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

• Akhmetev, P. M. ^[2] ; Medvedev, T. V. ^[1] ; Pochinka, O. V. ^[1]
  1. [1] Higher School of Economics, National Research University

     Higher School of Economics, National Research University

     Rusia

     
  2. [2] Pushkov Institute of Terrestrial Magnetism
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 3, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00518-1
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• Resumen

  • 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 π1(S2×S1). 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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