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Nelin. Dinam., 2019, Volume 15, Number 2, Pages 199–211 (Mi nd653)  

Mathematical problems of nonlinearity

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

T. V. Medvedeva, E. V. Nozdrinovab, O. V. Pochinkab, E. V. Shadrinab

a National Research University Higher School of Economics, ul. Rodionova 136, Niznhy Novgorod, 603093 Russia
b National Research University Higher School of Economics, ul. Bolshaya Pecherckaya 25/12, Niznhy Novgorod, 603155 Russia

Abstract: 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

Funding Agency Grant Number
Russian Science Foundation 17-11-01041
The construction of a stable arc (Theorem 2) is supported by RSF (Grant no. 17-11-01041), the splitting G into equivalence classes (Theorem 1) is supported by the Basic Research Program at the National Research University Higher School of Economics (HSE) in 2019.


DOI: https://doi.org/10.20537/nd190209

Full text: PDF file (534 kB)
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MSC: 37D15
Received: 05.06.2019
Accepted:20.06.2019

Citation: 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”, Nelin. Dinam., 15:2 (2019), 199–211

Citation in format AMSBIB
\Bibitem{MedNozPoc19}
\by T. V. Medvedev, E. V. Nozdrinova, O. V. Pochinka, E. V. Shadrina
\paper On a Class of Isotopic Connectivity of Gradient-like Maps of the 2-sphere with Saddles of Negative Orientation Type
\jour Nelin. Dinam.
\yr 2019
\vol 15
\issue 2
\pages 199--211
\mathnet{http://mi.mathnet.ru/nd653}
\crossref{https://doi.org/10.20537/nd190209}


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