Abstract
Structurally stable (rough) flows on surfaces have only finitely many singularities and finitely many closed orbits, all of which are hyperbolic, and they have no trajectories joining saddle points. The violation of the last property leads to \begin{document}$Ω$\end{document} -stable flows on surfaces, which are not structurally stable. However, in the present paper we prove that a topological classification of such flows is also reduced to a combinatorial problem. Our complete topological invariant is a multigraph, and we present a polynomial-time algorithm for the distinction of such graphs up to an isomorphism. We also present a graph criterion for orientability of the ambient manifold and a graph-associated formula for its Euler characteristic. Additionally, we give polynomial-time algorithms for checking the orientability and calculating the characteristic.
Highlights
A traditional method of qualitative studying of a flows dynamics with a finite number of special trajectories on surfaces consists of a splitting the ambient manifold by regions with a predictable trajectories behavior known as cells
We will say that Υφt is the equipped graph of φt and denote it by Υ∗φt if: (1) every E-vertex is equipped with the weight “+” or “−” in consistent and inconsistent case respectively; (2) every M-vertex is equipped with a four-colour graph ΓM corresponding to the flow f t constructed in Subsection 4.4; (3) every edge (M, L) ((L, M)) is equipped with an oriented tu-cycle τM,L of ΓM corresponding to the limit cycle c of L and oriented consistently with Rc
To solve the realization problem, we introduce the notions of an admissible four-colour graph and an equipped graph
Summary
A traditional method of qualitative studying of a flows dynamics with a finite number of special trajectories on surfaces consists of a splitting the ambient manifold by regions with a predictable trajectories behavior known as cells. Peixoto obtained a topological classification of structurally stable flows on arbitrary surfaces [22] As before, he did it by studying all admissible cells and he introduced a combinatorial invariant called a directed graph generalizing the Leontovich-Mayer scheme. Due to [18], the non-wandering set Ωφt of the flow φt consists of a finite number of hyperbolic fixed points and hyperbolic closed trajectories (limit cycles), which are called basic sets, denote them Ω1, . Denote by G a class of Ω-stable flows φt with at least one fixed saddle point or at least one limit cycle on a surface S. Let us extend φt|M to an Ω-stable flow f t : M → M such that f t coincides with φt out of D and Ωft has exactly one fixed point (a sink or a source) in each connected component of D. Denote by ∆ft the set of all polygonal regions of f t (see Fig. 5, where a flow f t and all its polygonal regions are presented)
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