
Mathematical problems of nonlinearity
On a Class of Isotopic Connectivity of Gradientlike Maps of the 2sphere with Saddles of Negative Orientation Type
T. V. Medvedev^{a}, E. V. Nozdrinova^{b}, O. V. Pochinka^{b}, E. V. Shadrina^{b} ^{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 gradientlike orientationpreserving diffeomorphisms
of the 2sphere 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
“sourcesink” diffeomorphism by a stable arc and this arc contains at most finitely many
points of perioddoubling bifurcations.
Keywords:
sinksource map, stable arc
Funding Agency 
Grant Number 
Russian Science Foundation 
171101041 
The construction of a stable arc (Theorem 2) is supported by RSF (Grant no. 171101041), 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
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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 Gradientlike Maps of the 2sphere 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 Gradientlike Maps of the 2sphere with Saddles of Negative Orientation Type
\jour Nelin. Dinam.
\yr 2019
\vol 15
\issue 2
\pages 199211
\mathnet{http://mi.mathnet.ru/nd653}
\crossref{https://doi.org/10.20537/nd190209}
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http://mi.mathnet.ru/eng/nd653 http://mi.mathnet.ru/eng/nd/v15/i2/p199
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