On the number of the classes of topological conjugacy of Pixton diffeomorphisms (Q1981917)

A knot on a manifold \(S^2\times S^1\) is a smooth embedding \(\gamma :S^1\rightarrow S^2\times S^1\) or its image. The generator of the fundamental group \(\pi _1(S^2\times S^1)\) is the Hopf knot equivalence class. For a wide class of dynamical systems known as Pixton diffeomorphisms, this equivalence class completely defines the topological conjugacy class. Any Hopf knot is realizable by a Pixton diffeomorphism. 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. The authors describe a first-order invariant for these knots. This result allows to model countable families of pairwise non-equivalent Hopf knots and an infinite set of topologically non-conjugate Pixton diffeomorphisms.