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Article

О числе модулей градиентных потоков функции высоты поверхности

In 1978 J. Palis invented continuum topologically non-conjugate systems in a neighbourhood of a system with a heteroclinic contact (moduli). W. de Melo and С. van Strien in 1987 described a diffeomorphism class with a finite number of moduli: 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 value 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.