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Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva, 2018, Volume 20, Number 4, Pages 419–428
DOI: https://doi.org/10.15507/2079-6900.20.201804.419-428
(Mi svmo718)
 

Mathematics

On number of moduli for gradient surface height function flows

V. E. Kruglov

National Research University – Higher School of Economics in Nizhny Novgorod
References:
Abstract: In 1978 J. Palis invented continuum topologically non-conjugate systems in a neighbourhood of a system with a heteroclinic contact; in other words, he invented so-called moduli. W. de Melo and С. van Strien in 1987 described a diffeomorphism class with a finite number of moduli. They discovered that a chain of saddles taking part in the heteroclinic contact of such diffeomorphism includes not more than three saddles. Surprisingly, such effect does not happen in flows. Here we consider gradient flows of the height function for an orientable surface of genus $g>0$. Such flows have a chain of $2g$ saddles. We found that the number of moduli for such flows is $2g-1$ which is the straight consequence of the sufficient topological conjugacy conditions for such systems given in our paper. A complete topological equivalence invariant for such systems is four-colour graph carrying the information about its cells relative position. Equipping the graph's edges with the analytical parameters — moduli, connected with the saddle connections, gives the sufficient conditions of the flows topological conjugacy.
Keywords: modulus of stability, gradient flow, topological conjugacy, four-colour graph, topological invariant.
Bibliographic databases:
Document Type: Article
UDC: 517.9
MSC: 34C20
Language: Russian
Citation: V. E. Kruglov, “On number of moduli for gradient surface height function flows”, Zhurnal SVMO, 20:4 (2018), 419–428
Citation in format AMSBIB
\Bibitem{Kru18}
\by V.~E.~Kruglov
\paper On number of moduli for gradient surface height function flows
\jour Zhurnal SVMO
\yr 2018
\vol 20
\issue 4
\pages 419--428
\mathnet{http://mi.mathnet.ru/svmo718}
\crossref{https://doi.org/10.15507/2079-6900.20.201804.419-428}
\elib{https://elibrary.ru/item.asp?id=37347612}
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    Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva
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