Geodesic
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, Tome 15 (2019) no. 2, pp. 199-211

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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
Mots-clés : stable arc. 
@article{ND_2019_15_2_a8,
     author = {T. V. Medvedev and E. V. Nozdrinova and O. V. Pochinka and E. V. Shadrina},
     title = {On a {Class} of {Isotopic} {Connectivity} of {Gradient-like} {Maps} of the 2-sphere with {Saddles} of {Negative} {Orientation} {Type}},
     journal = {Russian journal of nonlinear dynamics},
     pages = {199--211},
     publisher = {mathdoc},
     volume = {15},
     number = {2},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ND_2019_15_2_a8/}
}
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T. V. Medvedev; E. V. Nozdrinova; O. V. Pochinka; E. V. Shadrina. 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, Tome 15 (2019) no. 2, pp. 199-211. http://geodesic.mathdoc.fr/item/ND_2019_15_2_a8/