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# Topological classification of $Ω$-stable flows on surfaces by means of effectively distinguishable multigraphs

• * Corresponding author: Olga Pochinka

Authors are grateful to participants of the seminar "Topological Methods in Dynamics" for fruitful discussions. The classification results (Sections 1–6 without Subsections 5.2, 5.3) were obtained with the support of the Russian Science Foundation (project 17-11-01041). The realisation results (Subsection 5.2, Section 7) were obtained as an output of the research project "Topology and Chaos in Dynamics of Systems, Foliations and Deformation of Lie Algebras (2018)" implemented as part of the Basic Research Program at the National Research University Higher School of Economics (HSE). The algorithmic results (Subsection 5.3, Section 8) were obtained with the support of Russian Foundation for Basic Research 16-31-60008-mol-a-dk and with LATNA laboratory, National Research University Higher School of Economics

• 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 $Ω$-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.

Mathematics Subject Classification: 37D05.

 Citation:

• Figure 1.  The case when $U_\mathfrak c$ is homeomorphic to a Möbius band

Figure 2.  $\phi^t$ and $\Upsilon_{\phi^t}$

Figure 3.  The cases of the consistent (leftward) and the inconsistent (rightward) orientation of boundary's connecting component of some $\mathcal E$-region

Figure 4.  A polygonal region

Figure 5.  An example of the flow $f^t$ together with the polygonal regions

Figure 6.  An example of $f^t$ and its four-colour graph

Figure 7.  Two flows from $G$ and their equipped graphs

Figure 8.  Two examples of flows from $G$ differing only by orientation of the limit cycle between $\mathcal M$ and $\mathcal A$ and their equipped graphs

Figure 9.  Two examples of flow from $G$ without $\mathcal A$- and $\mathcal M$-regions differing only by orientation of the limit cycle and their equipped graphs

Figure 10.  $f^t$, $\Gamma_{\mathcal M}$ and $\Gamma^*_{{\mathcal M}}$

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