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Regular version of the site

Article

Construction of the Morse –Bott Energy Function for Regular Topological Flows

Regular and Chaotic Dynamics. 2021. Vol. 26. No. 4. P. 350-369.
Pochinka O., Zinina S.

In this paper, we consider regular topological flows on closed n-manifolds. Such
flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number
of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale
flows, which are closely related to the topology of the supporting manifold. This connection is
provided by the existence of the Morse –Bott energy function for the Morse – Smale flows. It
is well known that, starting from dimension 4, there exist nonsmoothing topological manifolds,
on which dynamical systems can be considered only in a continuous category. The existence of
continuous analogs of regular flows on any topological manifolds is an open question, as is the
existence of energy functions for such flows. In this paper, we study the dynamics of regular
topological flows, investigate the topology of the embedding and the asymptotic behavior of
invariant manifolds of fixed points and periodic orbits. The main result is the construction of
the Morse –Bott energy function for such flows, which ensures their close connection with the
topology of the ambient manifold.