On a simple isotopy class of gradient-like diffeomorphisms on a two-dimensional sphere

In 1976, S. Newhouse, J. Palis, F. Takens [2] introduced the concept of a stable arc connecting two structurally stable systems on a manifold. Such an arc does not change its qualitative properties under small perturbation. In the same year, Sh.Newhouse and M.Peixoto [3] proved the existence of a stable arc between any two Morse–Smale flows. For Morse–Smale diffeomorphisms on manifolds of any dimension, examples of systems that cannot be connected by a stable arc are known. This leads to an important question about finding an invariant that uniquely determines the equivalence class of a Morse–Smale diffeomorphism with respect to the connectedness relation by a stable arc (the stable connectedness component). So for orientation-preserving rough transformations of a circle, such an invariant is the Poincare rotation number [4]. For Morse-Smale diffeomorphisms on surfaces P. Blanchard [1] established certain necessary conditions for the existence of a stable arc connecting them. From these conditions, in particular, it follows that even on a two-dimensional sphere, there are infinitely many components of stable connectivity. In the present report necessary and sufficient conditions for the existence of a stable arc connecting gradient-like diffeomorphisms on a 2-dimensional sphere will be established.
Acknowledgments. This work was supported by the grant of the RSF 17-11-01041.
References
[1] Blanchard P.R., Invariant of the NPT isotopy classes of Morse–Smale diffeomorphisms of surfaces, Duke Mathematical Journal, 1980, vol 47, no. 1, pp.33–46.
[2] Newhouse S.E., Palis J., Takens F. Bifurcations and stability of families of diffeomorphisms. Publications mathematiques de l’ I.H.E.S, 57, 1983, pp.5–71.
[3] Newhouse, S., Peixoto, M.M. There is a simple arc joining any two Morse–Smale flows, Trois etudes en dynamique qualitative, Asterisque, 31, Soc. Math. France, Paris, 1976, pp. 15–41.
[4] Nozdrinova E. Rotation number as a complete topological invariant of a simple isotopic class of rough transformations of a circle // Russian Journal of Nonlinear Dynamics. 2018. Vol. 14. No. 4. pp. 543–551.