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## Topological conjugacy of gradient-like flows on surfaces

Динамические системы. 2018. Vol. 8(36). No. 1. P. 15-21.

The class of C^1-smooth gradient-like flows (Morse flows) on closed surface is the subclass
of the Morse-Smale flows class, which are rough. Their non-wandering set consists of a finite number
of hyperbolic fixed points and a finite number of hyperbolic limit cycles, and they does not have
trajectories connecting saddle points. It is well known that the topological equivalence class of a Morse-
Smale flow on a surface can be described combinatorially, for example, by the directed Peixoto graph,
or by the Oshemkov-Sharko molecule. However, the description of the class of the topological conjugacy
of such a system already requires the introduction of continuous invariants (moduli), corresponding
to the periods of limit cycles at least. Thus, one class of the equivalence contains continuum classes
of the topological conjugacy. Gradient-like flows are Morse-Smale flows without limit cycles. In this
paper we prove that gradient-like flows on a closed surface are topologically conjugate iff they are
topologically equivalent.