Topological conjugacy of gradient-like flows on $n$-dimensional sphere

Gradient-like flows are continuous dynamical systems whose non-wandering set consists of a finite number of hyperbolic fixed points. Their invariant manifolds cross each other transversally.
Depending on research goals there are two important things: qualitative behaviour of the system (i.e. partition of a manifold into trajectories) and moving along the trajectories by the time. In dynamical systems theory topological equivalence is an existence of a homeomorphism sending trajectories of one flows into trajectories of another one preserving direction of moving; if such a homeomorphism preserves time of moving along the trajectories, then it is called topological conjugacy of flows. Searching for invariant determining topological equivalence class for a system is topological classification.
The non-wandering set is a finite. Hence, the problem of topological classification may be reduced to a combinatorial one. First time it was done by E. Leontovich and A. Mayer in [2], [3] for classification of flows with finite number of singular trajectories on 2-dimensional sphere. These results were developed in researches by M.Peixoto [5], A.Oshemkov, V.Sharko [4], S.Pilyugin [6], A.Prishlyak [7], where similar problem was solved for Morse–Smale flows on closed manifolds of dimensions 2,3 and higher. These works were dedicated to topological equivalency. In [1] there is proved that topological equivalent flows on surfaces are also conjugate, hence, all equivalence results are also true for conjugacy. In our work we obtained similar result for class $G$ of gradient-like flows without heteroclinic trajectories on $n$-sphere, $n\geqslant3$. Besides, we introduce topological combinatorial invariant for such flows, i.e. bi-colour graph and prove that two flows from $G$ are topological conjugate iff their bi-colour graphs are isomorphic.
Acknowledgements. The work was done in collaboration with O.Pochinka and D.Malyshev with support of Russian Science Foundation, project No. 17-11-01041.
References
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