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Regular version of the site

Article

On a class of isotopic connectivity of gradient-like maps of the 2-sphere with saddles of negative orientation type

Russian Journal of Nonlinear Dynamics. 2019. Vol. 15. No. 2. P. 199-211.

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.