On a Class of Isotopic Connectivity of Gradient-like Maps of the 2-sphere with Saddles of Negative Orientation Type

    Received 05 June 2019; accepted 20 June 2019

    2019, Vol. 15, no. 2, pp.  199-211

    Author(s): Medvedev T. V., Nozdrinova E. V., Pochinka O. V., Shadrina E. V.

    We consider the class $G$ of gradient-like orientation-preserving diffeomorphisms of the 2-sphere with saddles of negative orientation type. We show that the for every diffeomorphism $f\in G$ every saddle point is fixed. We show that there are exactly three equivalence classes (up to topological conjugacy) $G=G_1\cup G_2\cup G_3$ where a diffeomorphism $f_1\in G_1$ has exactly one saddle and three nodes (one fixed source and two periodic sinks); a diffeomorphism $f_2\in G_2$ has exactly two saddles and four nodes (two periodic sources and two periodic sinks) and a diffeomorphism $f_3\in G_3$ is topologically conjugate to a diffeomorphism $f_1^{-1}$. The main result is the proof that every diffeomorphism $f\in G$ can be connected to the ``source-sink'' diffeomorphism by a stable arc and this arc contains at most finitely many points of period-doubling bifurcations.
    Keywords: sink-source map, stable arc
    Citation: Medvedev T. V., Nozdrinova E. V., Pochinka O. V., Shadrina E. V., On a Class of Isotopic Connectivity of Gradient-like Maps of the 2-sphere with Saddles of Negative Orientation Type, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 2, pp.  199-211
    DOI:10.20537/nd190209


    Download File
    PDF, 534.68 Kb

    References
    [1] Newhouse, S., Palis, J., and Takens, F., “Stable Arcs of Diffeomorphisms”, Bull. Amer. Math. Soc., 82:3 (1976), 499–502  crossref  mathscinet  zmath
    [2] Newhouse, S. and Peixoto, M. M., “There Is a Simple Arc Joining Any Two Morse – Smale Flows”, Trois études en dynamique qualitative, Astérisque, 31, Soc. Math. France, Paris, 1976, 15–41  mathscinet
    [3] Nozdrinova, E. V., “Rotation Number As a Complete Topological Invariant of a Simple Isotopic Class of Rough Transformations of a Circle”, Russian J. Nonlinear Dyn., 14:4 (2018), 543–551  mathscinet
    [4] Blanchard, P. R., “Invariants of the NPT Isotopy Classes of Morse – Smale Diffeomorphisms of Surfaces”, Duke Math. J., 47:1 (1980), 33–46  crossref  mathscinet  zmath
    [5] Grines, V., Medvedev, T., and Pochinka, O., Dynamical Systems on $2$- and $3$-Manifolds, Dev. Math., 46, Springer, New York, 2016, XXVI, 295 pp.  mathscinet  zmath
    [6] von Kerékjártó, B., “Über die periodischen Transformationen der Kreisscheibe und der Kugelflache”, Math. Ann., 80:1 (1919), 36–38  crossref  mathscinet  zmath
    [7] Newhouse, S., Palis, J., and Takens, F., “Bifurcations and Stability of Families of Diffeomorphisms”, Inst. Hautes Études Sci. Publ. Math., 1983, no. 57, 5–71  crossref  mathscinet  zmath
    [8] Milnor, J., Lectures on the $h$-Cobordism Theorem, Princeton Univ. Press, Princeton, N.J., 1965, v+116 pp.  mathscinet  zmath
    [9] Katok, A. and Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge, 1995, 802 pp.  mathscinet  zmath
    [10] Rolfsen, D., Knots and Links, Math. Lect. Ser., 7, Publish or Perish, Inc., Berkeley, Calif., 1976  mathscinet  zmath
    [11] Banyaga, A., “On the Structure of the Group of Equivariant Diffeomorphisms”, Topology, 16:3 (1977), 279–283  crossref  mathscinet  zmath



    Creative Commons License
    This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License