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We consider the class G of gradient-like orientation-preserving diffeomorphisms of the 2sphere with saddles of negative orientation type. We show that the for every diffeomorphism f ∈ G every saddle point is fixed. We show that there are exactly three equivalence classes (up to topological conjugacy) G = G1∪G2∪G3 where a diffeomorphism f1 ∈ G1 has exactly one saddle and three nodes (one fixed source and two periodic sinks); a diffeomorphism f2 ∈ G2 has exactly two saddles and four nodes (two periodic sources and two periodic sinks) and a diffeomorphism f3 ∈ G3 is topologically conjugate to a diffeomorphism f−1 1 . The main result is the proof that every diffeomorphism f ∈ 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.

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